#6692733

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667580
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	C₂×#449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1523 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)(17,18,24,23)(19,25,29,32)(20,22,26,21)(27,28,31,30);(1,3)(2,5)(4,8)(6,7)(9,11,13,12)(10,14,15,16)(17,19,24,29)(18,25,23,32)(20,30,26,28)(21,31,22,27);(9,12,13,11)(10,14,15,16)(17,20)(18,22)(19,27)(21,23)(24,26)(25,28)(29,31)(30,32);(2,6)(5,7)(17,21)(18,26)(19,27)(20,23)(22,24)(25,30)(28,32)(29,31);(3,8)(5,7)(17,22,24,21)(18,26,23,20)(19,28,29,30)(25,31,32,27)

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=g,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,d^b=di,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, 6599929);