#6840635

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	#25133	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1492 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)(11,14)(12,13)(15,16)(17,18)(19,21)(20,29)(22,31)(23,26)(24,25)(27,30)(28,32);(1,2)(3,6)(4,5)(7,8)(10,13)(11,15)(17,19,23,25)(18,24,26,21)(20,22,31,29)(27,28,32,30);(10,13)(14,16)(17,20,21,28,23,31,24,30)(18,22,19,27,26,29,25,32);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(17,21,23,24)(18,25,26,19)(20,30,31,28)(22,27,29,32);(2,5)(6,8)(9,11,12,15)(10,14,13,16)(17,22,21,27,23,29,24,32)(18,20,19,28,26,31,25,30)

Polycyclic presentation:

2^9|b^2=i,c^2=hi,d^2=fgi,e^2=gh,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, 6747831);