#7502115

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×#3062	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2695 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)(10,15)(12,22)(13,24)(16,27)(17,30)(18,26)(19,32)(21,34)(25,31)(28,38)(29,36)(35,39);(5,6)(7,8)(9,10,14,18)(11,16,20,29)(12,23,25,37)(13,26,24,15)(17,31,30,22)(19,33,35,40)(21,36,34,27)(28,39,38,32);(2,4)(6,8)(9,11)(10,16)(12,19)(13,21)(14,20)(15,27)(17,28)(18,29)(22,32)(23,33)(24,34)(25,35)(26,36)(30,38)(31,39)(37,40);(2,4)(9,12,24,31,14,25,13,22)(10,17,26,37,18,30,15,23)(11,19,34,39,20,35,21,32)(16,28,36,40,29,38,27,33);(11,20)(16,29)(19,35)(21,34)(27,36)(28,38)(32,39)(33,40)

Polycyclic presentation:

2^9|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, 7409311);