#7514899

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514160
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	#23908	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1889 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)(3,6)(4,5)(7,8)(9,10,16,21)(11,17,23,19)(12,24,27,36)(13,28,29,33)(14,20,15,30)(18,31,37,26)(22,38,40,34)(25,35,39,32);(2,5)(6,8)(9,11,16,23)(10,17,21,19)(12,25,27,39)(13,15,29,14)(18,34,37,38)(20,28,30,33)(22,26,40,31)(24,35,36,32);(1,3,4,7)(2,6,5,8)(9,12)(10,18)(11,22)(13,25)(14,26)(15,31)(16,27)(17,32)(19,35)(20,36)(21,37)(23,40)(24,30)(28,38)(29,39)(33,34);(9,13,16,29)(10,19,21,17)(11,14,23,15)(12,25,27,39)(18,35,37,32)(20,33,30,28)(22,26,40,31)(24,38,36,34);(2,5)(6,8)(9,14,16,15)(10,20,21,30)(11,13,23,29)(12,26,27,31)(17,28,19,33)(18,36,37,24)(22,25,40,39)(32,38,35,34)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=f,d^2=i,e^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,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, 7422095);