#7613492

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 ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1876 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)(3,6)(4,5)(7,8)(9,10)(11,16)(12,23)(13,26)(14,31)(15,19)(17,30)(18,24)(20,37)(21,36)(22,33)(25,34)(27,39)(28,32)(29,40)(35,38);(2,5)(6,8)(9,11,15,22)(10,16,19,33)(12,24,21,14)(13,27,29,20)(17,32,35,34)(18,36,31,23)(25,30,28,38)(26,39,40,37);(1,3,4,7)(2,6,5,8)(9,12,14,11,15,21,24,22)(10,16,18,23,19,33,31,36)(13,28,30,27,29,25,38,20)(17,34,26,39,35,32,40,37);(9,13)(10,17,19,35)(11,20)(12,25)(14,30)(15,29)(16,32,33,34)(18,26,31,40)(21,28)(22,27)(23,37,36,39)(24,38);(2,5)(6,8)(13,29)(17,35)(20,27)(25,28)(26,40)(30,38)(32,34)(37,39)

Polycyclic presentation:

2^9|b^2=i,c^2=fg,d^2=hi,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, 7520688);