#7156965

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996355
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

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 ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1975 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,c^2=fg,d^2=fi,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dhi,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72280990, #72281161, #72281290
maximal normal subgroup of	#72280990, #72281161, #72281290

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7064161);