#6812806

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806839
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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1333 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=hi,d^2=fi,e^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^c=ei,e^d=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, 6720002);