#6812725

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806671
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	#42658	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1341 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)(17,18,22,23)(19,24,26,21)(20,28,30,25)(27,31,32,29);(1,2)(3,6)(4,5)(7,8)(9,11,12,15)(10,13,14,16)(17,19,22,26)(18,24,23,21)(20,29,30,31)(25,32,28,27);(10,14)(13,16)(17,20,18,25,22,30,23,28)(19,27,21,31,26,32,24,29);(1,3,4,7)(2,6,5,8)(20,30)(25,28)(27,32)(29,31);(2,5)(6,8)(17,21,23,19,22,24,18,26)(20,31,28,27,30,29,25,32)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6719921);