#7283545

order	512
nilpotency class	3
isoclinism	isoclinic class of #7234345
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1680 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=i,d^2=h,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7190741);