#7500981

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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3240 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)(11,16)(12,24)(13,19)(14,29)(15,20)(17,26)(18,22)(21,33)(23,32)(25,34)(27,35)(28,31)(30,36)(37,39)(38,40);(1,2)(3,4)(5,6)(7,8)(9,11)(10,16)(12,21)(13,28)(14,22)(15,23)(17,30)(18,29)(19,31)(20,32)(24,33)(25,38)(26,36)(27,37)(34,40)(35,39);(6,8)(9,12,15,27)(10,17,20,34)(11,21,23,37)(13,26,14,25)(16,30,32,40)(18,33,31,39)(19,35,29,24)(22,38,28,36);(2,4)(9,11)(10,18)(12,21)(13,22)(14,28)(15,23)(16,29)(17,33)(19,32)(20,31)(24,30)(25,36)(26,38)(27,37)(34,39)(35,40);(9,13,15,14)(10,19,20,29)(11,22,23,28)(12,25,27,26)(16,18,32,31)(17,24,34,35)(21,36,37,38)(30,39,40,33)

Polycyclic presentation:

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

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, 7408177);