#7622590

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

1954 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)(5,6,10,15,11,7,16,19,9,12,23,26,24,17,13,27)(8,14,21,29,22,18,30,32,20,25,33,35,34,31,28,36);(2,4)(6,12)(7,17)(14,25)(15,26)(18,31)(19,27)(29,35)(32,36);(5,7,10,19,11,12,16,26,9,17,23,27,24,6,13,15)(8,18,21,32,22,25,30,35,20,31,33,36,34,14,28,29);(5,8)(6,14)(7,18)(9,20)(10,21)(11,22)(12,25)(13,28)(15,29)(16,30)(17,31)(19,32)(23,33)(24,34)(26,35)(27,36);(8,20)(14,25)(18,31)(21,33)(22,34)(28,30)(29,35)(32,36)

Polycyclic presentation:

2^9|a^2=g,c^2=g,g^2=h,h^2=i,b^a=bf,c^a=ci,c^b=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, 7529786);