#7514917

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514172
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	#61187	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1874 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)(11,20)(12,23)(13,26)(14,18)(15,21)(16,25)(17,28)(19,29)(22,37)(24,33)(27,36)(30,31)(32,35)(34,39)(38,40);(2,5)(6,8)(9,11,14,23)(10,12,18,20)(13,22,27,38)(15,30,29,16)(17,34,35,24)(19,31,21,25)(26,40,36,37)(28,33,32,39);(1,3,4,7)(2,6,5,8)(9,12,14,20)(10,16,18,30)(11,21,23,19)(13,24,27,34)(15,31,29,25)(17,33,35,39)(22,26,38,36)(28,37,32,40);(9,13,14,27)(10,17,18,35)(11,22,23,38)(12,24,20,34)(15,32,29,28)(16,33,30,39)(19,36,21,26)(25,37,31,40);(2,5)(6,8)(10,18)(11,23)(16,30)(17,35)(19,21)(22,38)(26,36)(33,39)

Polycyclic presentation:

2^9|b^2=i,c^2=f,d^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cgi,d^b=dh,e^b=ei,g^b=gi,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, 7422113);