#233538

order	512
nilpotency class	3
isoclinism	isoclinic class of #10336
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#392
commutator, Frattini, cosocle	C₂⁵	C₂⁴
socle	C₂²	C₂×#392
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

2636 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|d^2=h,b^a=be,c^a=cf,g^a=ghi,c^b=cg,f^b=fi,e^c=eh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72232871, #72234080, #72234091, …
maximal normal subgroup of	#72232871, #72234080, #72234091, …
maximal quotient of	#72232859, #72233046, C₄×#10336, …
outer automorphism group of	C₂×C₂²⋊Dic₅

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 140734);