#7500841

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

3288 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)(9,10,14,19,15,20,13,16)(11,17,22,31,23,32,27,29)(12,24,25,33,26,34,28,18)(21,35,36,39,38,40,37,30);(5,6)(7,8)(10,16)(12,25)(13,14)(17,29)(18,33)(19,20)(21,36)(22,27)(26,28)(30,39)(31,32)(37,38);(2,4)(6,8)(9,11,15,23)(10,17,20,32)(12,21,26,38)(13,27,14,22)(16,29,19,31)(18,30,33,39)(24,35,34,40)(25,36,28,37);(2,4)(9,12)(10,18)(11,21)(13,25)(14,28)(15,26)(16,24)(17,30)(19,34)(20,33)(22,37)(23,38)(27,36)(29,35)(31,40)(32,39);(9,13,15,14)(10,16,20,19)(11,22,23,27)(12,25,26,28)(17,31,32,29)(18,24,33,34)(21,37,38,36)(30,40,39,35)

Polycyclic presentation:

2^9|a^2=f,c^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, 7408037);