#7336517

order	512
nilpotency class	3
isoclinism	isoclinic class of #7324564
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1491 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=g,d^2=f,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,d^c=dhi,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, 7243713);