#7576752

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514040
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×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1353 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)(17,18,22,26)(19,23,30,37)(20,24,31,38)(21,25,33,40)(27,34,42,46)(28,35,29,36)(32,39,44,48)(41,45,43,47);(2,6)(7,8)(10,13)(11,16)(17,19,22,30)(18,23,26,37)(20,27,31,42)(21,28,33,29)(24,34,38,46)(25,35,40,36)(32,41,44,43)(39,45,48,47);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,14,13,15)(17,20,22,31)(18,24,26,38)(19,27,30,42)(21,32,33,44)(23,34,37,46)(25,39,40,48)(28,41,29,43)(35,45,36,47);(10,13)(14,15)(19,28)(21,33)(23,35)(25,40)(27,41)(29,30)(32,44)(34,45)(36,37)(39,48)(42,43)(46,47);(1,4,5,3)(2,7,6,8)(10,13)(14,15)(20,31)(24,38)(27,42)(32,44)(34,46)(39,48)(41,43)(45,47)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fi,e^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,e^c=ei,h^d=hi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7483948);