#72268637

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₄×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#1270
commutator	C₂×C₄²	C₄×C₈
Frattini, cosocle	C₂×C₄×#95	C₂²
prime radical	C₂⁵	#99
LCS	G ▷ C₂×C₄² ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ C₂×C₄×#95 ◁ G
LECS	G ▷ C₂×C₄×#95 ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1465 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 2 generators on 28 points:

(5,6,7,10)(8,9,11,12)(13,14)(15,22,23,28,25,27,24,18)(16,20)(17,21)(19,26);(1,2,3,4)(6,9)(13,15)(14,18)(16,23)(17,25)(19,27)(20,22)(21,28)(24,26)

Polycyclic presentation:

2^10|a^2=c,b^2=e,c^2=g,d^2=hj,f^2=j,b^a=bd,c^b=cf,d^a=dfh,d^b=dhj,d^c=di,f^a=fi,f^b=fj,g^b=gj,h^a=hj

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72268639
maximal normal subgroup of	#72268639

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.3);;
gap> SetPower(coll, 2, F.5);;
gap> SetPower(coll, 3, F.7);;
gap> SetPower(coll, 4, F.8*F.10);;
gap> SetPower(coll, 6, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.6*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.8*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);