#72258606

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#15948
commutator, Frattini, cosocle	C₂⁶	C₂⁴
prime radical	#6499870	C₂
LCS	G ▷ C₂⁶ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁶ ▷ C₂² ▷ C₁

5397 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 24 points:

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

Polycyclic presentation:

2^10|a^2=f,b^2=i,c^2=h,b^a=be,c^a=cefij,c^b=cfg,d^a=dfgh,d^b=dfgi,d^c=dhi,e^a=ej,e^c=ej,e^d=ei,f^b=fj,f^c=fj,g^b=gj,g^c=gij,h^b=hi

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72258632, #72258757, #72303433, …
maximal normal subgroup of	#72258632, #72258757, #72303433, …

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