#6837059

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806903
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	#39881	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#121 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1685 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:

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

Polycyclic presentation:

2^9|b^2=fi,c^2=ghi,d^2=fi,e^2=gh,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei,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, 6744255);