#7168183

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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
prime radical	#42478	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1654 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,d^2=f,e^2=h,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=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7075379);