#6740430

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	#42570	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1483 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:

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

Polycyclic presentation:

2^9|b^2=fi,c^2=f,d^2=gi,e^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,e^c=ei,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, 6647626);