#6877367

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682360
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	#30400	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1437 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|b^2=i,c^2=fgi,d^2=hi,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,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, 6784563);