#7502128

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

2630 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)(9,10,14,17)(11,19,22,35)(12,23,27,34)(13,28,25,18)(15,29,31,21)(16,26,33,36)(20,30,37,38)(24,39,40,32);(5,6)(7,8)(9,11,14,22)(10,15,17,31)(12,24,27,40)(13,29,25,21)(16,32,33,39)(18,35,28,19)(20,36,37,26)(23,38,34,30);(2,4)(6,8)(9,10)(11,15)(12,16)(13,18)(14,17)(19,21)(20,30)(22,31)(23,36)(24,32)(25,28)(26,34)(27,33)(29,35)(37,38)(39,40);(2,4)(9,12,25,36,14,27,13,26)(10,16,28,23,17,33,18,34)(11,20,29,40,22,37,21,24)(15,30,35,39,31,38,19,32);(10,17)(15,31)(16,33)(18,28)(19,35)(23,34)(30,38)(32,39)

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,e^a=ei,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, 7409324);