#7503722

order	512
nilpotency class	3
isoclinism	isoclinic class of #7501009
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1940 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)(9,10,14,19,15,20,16,21)(11,22,25,31,18,13,24,32)(12,26,33,17,30,37,29,36)(23,34,27,38,35,40,39,28);(1,3,5,2)(4,7,8,6)(9,11)(10,13)(12,27)(14,25)(15,18)(16,24)(17,34)(19,32)(20,22)(21,31)(23,29)(26,28)(30,39)(33,35)(36,40)(37,38);(1,4,5,8)(2,6,3,7)(9,12,15,30)(10,17,20,36)(11,23,18,35)(13,28,22,38)(14,29,16,33)(19,26,21,37)(24,27,25,39)(31,34,32,40);(2,3)(6,7)(9,13)(10,18)(11,20)(12,28)(14,31)(15,22)(16,32)(17,35)(19,25)(21,24)(23,36)(26,39)(27,37)(29,34)(30,38)(33,40);(9,14,15,16)(10,19,20,21)(11,24,18,25)(12,29,30,33)(13,31,22,32)(17,26,36,37)(23,27,35,39)(28,34,38,40)

Polycyclic presentation:

2^9|a^2=gi,b^2=fi,c^2=f,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,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, 7410918);