#6720163

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
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	#25376	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1485 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=gi,c^2=fi,d^2=fi,e^2=gi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efhi,h^b=hi,h^c=hi,e^d=ei,h^e=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6627359);