#6825684

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806647
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×#496	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1367 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:

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

Polycyclic presentation:

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

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, 6732880);