#7502116

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500669
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	C₂×#537	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2625 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=fi,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^c=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, 7409312);