#6677852

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667460
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#120 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2966 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2)(3,7)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18,21,24)(19,22,27,31)(20,23,28,32)(25,29,26,30);(1,3)(2,5)(4,8)(6,7)(9,11,12,15)(10,13,14,16)(17,19,21,27)(18,22,24,31)(20,25,28,26)(23,29,32,30);(10,14)(13,16)(19,25)(20,28)(22,29)(23,32)(26,27)(30,31);(2,6)(5,7);(3,8)(5,7)(19,25)(20,28)(22,29)(23,32)(26,27)(30,31)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,h^c=hi,h^e=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239589, #72256920, #72256986
maximal normal subgroup of	#72239589, #72256920, #72256986
maximal quotient of	#72239786, #72265143

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6585048);