#7520548

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514148
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1747 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=f,d^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,g^c=gi,h^c=hi,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, 7427744);