#7598064

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518964
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	C₂×#548	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1461 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)(3,7)(4,8)(5,6)(9,10)(11,15)(12,13)(14,16)(17,18)(19,26)(20,32)(21,36)(22,27)(23,39)(24,30)(25,35)(28,37)(29,45)(31,41)(33,44)(34,48)(38,40)(42,47)(43,46);(2,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19,22,30)(18,24,27,26)(20,33,34,38)(21,37,31,23)(25,42,43,45)(28,36,39,41)(29,46,47,35)(32,40,48,44);(10,13)(14,15)(17,20,23,35,22,34,37,46)(18,25,28,32,27,43,39,48)(19,29,31,33,30,47,21,38)(24,40,41,42,26,44,36,45);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,14,13,15)(17,21,23,19,22,31,37,30)(18,26,28,36,27,24,39,41)(20,33,35,47,34,38,46,29)(25,40,32,42,43,44,48,45);(1,4,5,3)(2,7,6,8)(18,27)(20,34)(24,26)(28,39)(29,47)(33,38)(35,46)(36,41)

Polycyclic presentation:

2^9|b^2=i,c^2=h,d^2=fgi,e^2=fi,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^c=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, 7505260);