#6816580

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806647
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#553	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1728 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 36 points:

(2,4)(5,6)(7,14)(8,17)(9,12)(10,13)(11,16)(15,19)(18,20)(21,22)(23,24)(25,27)(26,29)(28,32)(30,35)(31,33)(34,36);(5,7)(6,11)(8,15)(9,18)(10,16)(12,19)(13,14)(17,20)(22,27)(23,30)(24,34)(26,31)(28,35)(32,36);(1,2,3,4)(21,23,31,36,25,32,26,30)(22,28,33,35,27,24,29,34);(5,8,10,9)(6,12,13,17)(7,15,16,18)(11,19,14,20)(22,27)(23,32)(29,33)(30,36);(5,9,10,8)(6,12,13,17)(7,15,16,18)(11,20,14,19)(21,24,31,34,25,28,26,35)(22,23,33,36,27,32,29,30)

Polycyclic presentation:

2^9|c^2=gh,d^2=fi,e^2=fh,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

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, 6723776);