#6684031

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
prime radical	#2414	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1775 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=gi,d^2=fi,e^2=i,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, 6591227);