#6686629

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	#2540	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1716 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)(8,17)(9,12)(10,13)(11,16)(15,19)(18,20)(21,22,26,27)(23,28,32,36)(24,33,31,29)(25,30,35,34);(5,7)(6,11)(8,15)(9,18)(10,16)(12,19)(13,14)(17,20)(21,23,26,32)(22,28,27,36)(24,30,31,34)(25,33,35,29);(1,2,3,4)(21,24)(22,29)(23,25)(26,31)(27,33)(28,34)(30,36)(32,35);(5,8,10,9)(6,12,13,17)(7,15,16,18)(11,19,14,20)(21,22,26,27)(23,28,32,36)(24,29,31,33)(25,34,35,30);(5,9,10,8)(6,12,13,17)(7,15,16,18)(11,20,14,19)(21,25,26,35)(22,30,27,34)(23,31,32,24)(28,29,36,33)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=gi,d^2=fi,e^2=fi,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, 6593825);