#6834626

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806647
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	#42462	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1655 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=ghi,e^2=ghi,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,e^c=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, 6741822);