#6675560

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
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	#42794	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1805 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:

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

Polycyclic presentation:

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

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, 6582756);