#6378755

order	512
nilpotency class	2
isoclinism	isoclinic class of #2668
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₈	C₂⁴
commutator, socle	C₂³	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#392	C₂×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1446 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)(17,18,21,25,22,20,24,29)(19,23,28,31,26,27,30,32);(1,2,4,5)(3,6,7,8)(9,11,12,15)(10,13,14,16)(17,19)(18,23)(20,27)(21,28)(22,26)(24,30)(25,31)(29,32);(2,5)(6,8)(10,14)(13,16)(17,19)(18,23)(20,27)(21,28)(22,26)(24,30)(25,31)(29,32);(1,3,4,7)(2,6,5,8)(19,26)(23,27)(28,30)(31,32);(17,20,21,29,22,18,24,25)(19,27,28,32,26,23,30,31)

Polycyclic presentation:

2^9|a^2=h,b^2=fg,d^2=f,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=cf,d^b=di,d^c=di

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6285951);