#6684321

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667556
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1747 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)(8,12)(9,18)(10,13)(11,17)(15,20)(16,19)(21,22)(23,25)(24,34)(26,29)(27,28)(30,33)(31,36)(32,35);(5,7)(6,11)(8,15)(9,16)(10,17)(12,19)(13,14)(18,20)(21,22,26,29)(23,30,32,36)(24,28,27,34)(25,31,35,33);(1,2,3,4)(5,8,10,9)(6,12,13,18)(7,15,17,16)(11,19,14,20)(21,23,26,32)(22,25,29,35)(24,33,27,31)(28,36,34,30);(5,9,10,8)(6,12,13,18)(7,16,17,15)(11,19,14,20)(21,24,26,27)(22,28,29,34)(23,31,32,33)(25,30,35,36);(7,17)(11,14)(15,16)(19,20)(21,25,26,35)(22,23,29,32)(24,30,27,36)(28,31,34,33)

Polycyclic presentation:

2^9|b^2=i,c^2=fg,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6591517);