#7614273

order	512
nilpotency class	3
isoclinism	isoclinic class of #7563333
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

1745 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)(3,7)(4,8)(5,6)(9,10)(11,19)(12,24)(13,28)(14,18)(15,20)(16,27)(17,21)(22,32)(23,34)(25,29)(26,39)(30,36)(31,37)(33,35)(38,40);(2,6)(7,8)(11,20)(13,21)(15,19)(16,31)(17,28)(22,35)(23,36)(25,40)(27,37)(29,38)(30,34)(32,33);(1,3,5,4)(2,7,6,8)(9,11,21,36,14,23,13,20)(10,15,28,34,18,30,17,19)(12,25,37,35,26,22,27,40)(16,32,39,33,31,29,24,38);(9,12,14,26)(10,16)(11,22,23,25)(13,27,21,37)(15,29)(17,24)(18,31)(19,33)(20,35,36,40)(28,39)(30,32)(34,38);(1,4,5,3)(2,7,6,8)(12,26)(16,31)(22,25)(24,39)(27,37)(29,32)(33,38)(35,40)

Polycyclic presentation:

2^9|c^2=fgi,d^2=hi,e^2=f,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,d^c=di,e^d=ei

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, 7521469);