#7621080

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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1285 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)(11,14)(12,13)(15,16)(17,18,24,30)(19,25,33,41)(20,26,36,44)(21,27,32,40)(22,28,23,29)(31,39,47,48)(34,42,37,45)(35,43,38,46);(2,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19,24,33)(18,25,30,41)(20,31,36,47)(21,22,32,23)(26,39,44,48)(27,28,40,29)(34,35,37,38)(42,43,45,46);(1,3,5,4)(2,7,6,8)(10,13)(14,16)(17,20,24,36)(18,26,30,44)(19,31,33,47)(21,37,32,34)(22,38,23,35)(25,39,41,48)(27,45,40,42)(28,46,29,43);(10,13)(14,16)(17,21,23,19,24,32,22,33)(18,27,29,25,30,40,28,41)(20,34,35,47,36,37,38,31)(26,42,43,48,44,45,46,39);(1,4,5,3)(2,7,6,8)(9,11,12,15)(10,14,13,16)(17,22,24,23)(18,28,30,29)(19,32,33,21)(20,35,36,38)(25,40,41,27)(26,43,44,46)(31,37,47,34)(39,45,48,42)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fi,d^2=gh,e^2=fgi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei,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, 7528276);