#6815607

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682360
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1968 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=fgh,d^2=f,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,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, 6722803);