#7500990

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500669
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3002	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3249 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)(9,10,14,19,15,20,13,18)(11,21,24,33,25,32,23,16)(12,26,28,35,29,36,30,17)(22,31,27,34,38,40,39,37);(5,6)(7,8)(9,11,15,25)(10,16,20,33)(12,27,29,39)(13,24,14,23)(17,34,35,37)(18,21,19,32)(22,30,38,28)(26,31,36,40);(2,4)(6,8);(2,4)(9,12)(10,17)(11,22)(13,28)(14,30)(15,29)(16,31)(18,26)(19,36)(20,35)(21,37)(23,27)(24,39)(25,38)(32,34)(33,40);(9,13,15,14)(10,18,20,19)(11,23,25,24)(12,28,29,30)(16,32,33,21)(17,26,35,36)(22,27,38,39)(31,34,40,37)

Polycyclic presentation:

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

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, 7408186);