#6815781

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806839
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×#3319	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1825 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,c^2=fghi,d^2=f,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,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, 6722977);