#7502136

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500701
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#23386	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2686 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

(1,2)(3,4)(9,10)(11,20)(12,24)(13,18)(14,29)(15,19)(16,22)(17,26)(21,37)(23,32)(25,34)(27,35)(28,36)(30,33)(31,39)(38,40);(5,6)(7,8)(9,11)(10,16)(12,25)(13,22)(14,30)(15,23)(17,34)(18,32)(19,33)(20,29)(21,26)(24,31)(27,38)(28,39)(35,37)(36,40);(2,4)(6,8);(2,4)(9,12,14,26,15,27,13,28)(10,17,18,24,19,36,29,35)(11,21,22,25,23,39,30,38)(16,31,20,34,33,37,32,40);(9,13,15,14)(10,18,19,29)(11,22,23,30)(12,26,27,28)(16,32,33,20)(17,35,36,24)(21,38,39,25)(31,34,37,40)

Polycyclic presentation:

2^9|d^2=f,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=dfg,f^a=fi,c^b=ch,d^b=df,f^b=fi,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7409332);