#6718178

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	#25087	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1610 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)(3,7)(4,6)(5,8)(9,10,13,15)(11,14,12,16)(17,18,23,20)(19,24,29,30)(21,31,22,25)(26,28,32,27);(1,3)(2,5)(4,8)(6,7)(9,11,13,12)(10,14,15,16)(17,19,23,29)(18,24,20,30)(21,27,22,28)(25,32,31,26);(10,15)(14,16)(17,20,23,18)(19,26,29,32)(21,31,22,25)(24,28,30,27);(2,6)(5,7)(17,21,23,22)(18,25,20,31)(19,27,29,28)(24,32,30,26);(3,8)(5,7)(9,12,13,11)(10,16,15,14)(19,28)(21,22)(24,32)(25,31)(26,30)(27,29)

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=i,d^2=i,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,e^b=efh,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, 6625374);