#7501831

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500736
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2330 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7409027);