#7500839

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500689
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×#3008	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3304 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)(9,10,14,20,15,21,13,19)(11,22,26,33,27,32,25,16)(12,17,28,34,29,35,30,18)(23,36,38,40,39,37,24,31);(5,6)(7,8)(9,11,15,27)(10,16,21,33)(12,23,29,39)(13,26,14,25)(17,31,35,40)(18,36,34,37)(19,22,20,32)(24,30,38,28);(2,4)(6,8)(9,12)(10,17)(11,23)(13,30)(14,28)(15,29)(16,31)(18,19)(20,34)(21,35)(22,36)(24,25)(26,38)(27,39)(32,37)(33,40);(2,4)(9,12)(10,18)(11,24)(13,28)(14,30)(15,29)(16,31)(17,19)(20,35)(21,34)(22,37)(23,25)(26,39)(27,38)(32,36)(33,40);(9,13,15,14)(10,19,21,20)(11,25,27,26)(12,28,29,30)(16,32,33,22)(17,34,35,18)(23,38,39,24)(31,36,40,37)

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,e^c=ei,f^d=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7408035);