#7622591

order	512
nilpotency class	2
isoclinism	isoclinic class of #3484
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₈ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂×C₈	C₂⁵
commutator, socle	C₂²	C₂⁴×C₈
prime radical	D₄×#41	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1961 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=g,g^2=h,h^2=i,b^a=bf,c^a=ci,e^d=ei

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7529787);