#7603870

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1379 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)(4,5)(6,8)(9,10,13,14)(11,16,12,15)(17,18)(19,27)(20,31)(21,23)(22,36)(24,28)(25,32)(26,35)(29,42)(30,39)(33,40)(34,41)(37,45)(38,46)(43,44)(47,48);(2,5)(6,7)(9,10)(11,15)(12,16)(13,14)(18,23)(19,28)(20,32)(22,35)(24,39)(25,41)(27,37)(29,33)(30,45)(31,38)(34,46)(43,47);(10,14)(15,16)(17,19,22,28,21,30,35,45)(18,24,26,27,23,37,36,39)(20,33,32,48,34,29,46,44)(25,42,31,47,38,40,41,43);(9,11,13,12)(10,15,14,16)(17,20,22,32,21,34,35,46)(18,25,26,31,23,38,36,41)(19,29,28,44,30,33,45,48)(24,40,27,43,37,42,39,47);(1,3,4,8)(2,6,5,7)(9,12,13,11)(10,16,14,15)(18,23)(19,30)(25,38)(26,36)(28,45)(29,33)(31,41)(44,48)

Polycyclic presentation:

2^9|a^2=ghi,c^2=h,d^2=g,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, 7511066);