#6687164

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667604
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 ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1953 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)(7,15)(8,12)(9,13)(10,14)(11,17)(16,20)(18,19)(22,28)(23,33)(24,25)(32,35);(5,7)(6,11)(8,16)(9,18)(10,17)(12,19)(13,20)(14,15)(21,22)(23,31)(24,30)(25,29)(26,28)(27,35)(32,34)(33,36);(1,2,3,4)(5,8,10,9)(6,12,14,13)(7,16,17,18)(11,19,15,20)(21,23)(22,29)(24,27)(25,34)(26,33)(28,30)(31,35)(32,36);(5,8,10,9)(6,13,14,12)(7,16,17,18)(11,20,15,19)(21,24,26,25)(22,30,28,29)(23,34,33,27)(31,32,36,35);(5,9,10,8)(6,13,14,12)(7,16,17,18)(11,19,15,20)(21,25,26,24)(22,31,28,36)(23,34,33,27)(29,35,30,32)

Polycyclic presentation:

2^9|c^2=fg,d^2=f,e^2=f,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,h^c=hi,h^e=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6594360);