#72279271

order	1024
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#1390
commutator	C₂×C₄⋊C₄	C₂×C₄²
Frattini, cosocle	C₂×#390	C₂³
prime radical	C₂²×#127	D₄
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×C₄² ◁ C₂×C₄×#127 ◁ G
LECS	G ▷ C₂×#390 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

1668 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 24 points:

(2,5,6,7)(9,10)(11,18)(12,16)(13,15)(14,19)(17,21)(20,22)(23,24);(1,2)(3,7)(4,6)(5,8)(9,11)(10,14,16,18)(12,19,21,24)(13,20)(15,23,17,22);(10,15)(11,19)(12,21)(14,22)(18,23)(20,24)

Polycyclic presentation:

2^10|a^2=d,b^2=e,f^2=h,g^2=j,b^a=bf,c^a=cj,c^b=cgj,d^b=dh,e^a=eg,e^c=eij,f^b=fghj,f^e=fij,g^a=gj,g^b=gi,g^c=gj,g^e=gj,g^f=gj,i^a=ij,i^b=ij

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72279309
maximal normal subgroup of	#72279309
maximal quotient of	#72279985

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 2, F.5);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.7*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 5, 3, F.5*F.9*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.7*F.8*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.9*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);