#7602899

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519008
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1345 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2)(3,6)(4,5)(7,8)(9,10,12,14)(11,15,16,13)(17,18)(19,28)(20,31)(21,34)(22,26)(23,39)(24,30)(25,37)(27,38)(29,45)(32,42)(33,44)(35,40)(36,48)(41,47)(43,46);(2,5)(6,8)(9,11)(10,13)(12,16)(14,15)(17,19,22,30)(18,24,26,28)(20,23,33,38)(21,32,36,35)(25,41,43,45)(27,44,39,31)(29,46,47,37)(34,40,48,42);(10,14)(11,16)(17,20,23,30,22,33,38,19)(18,24,27,31,26,28,39,44)(21,35,37,29,36,32,46,47)(25,42,34,45,43,40,48,41);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(17,21,23,37,22,36,38,46)(18,25,27,34,26,43,39,48)(19,29,20,32,30,47,33,35)(24,40,31,41,28,42,44,45);(2,5)(6,8)(10,14)(13,15)(18,26)(21,36)(24,28)(27,39)(29,47)(31,44)(32,35)(37,46)

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,c^2=h,d^2=fgi,g^2=i,h^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,h^b=hi,d^c=di,e^d=ei

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, 7510095);