#7610807

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	C₂×#553	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1470 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)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18,21,23)(19,24,28,35)(20,30,33,25)(22,26,34,36)(27,40,32,37)(29,38,43,46)(31,39,45,41)(42,47,44,48);(1,3,5,4)(2,6,8,7)(9,11)(10,13)(12,15)(14,16)(18,23)(20,31)(22,34)(24,35)(25,39)(27,42)(29,43)(30,41)(32,44)(33,45)(37,47)(40,48);(1,3,5,4)(2,7,8,6)(10,14)(11,15)(17,19,21,28)(18,24,23,35)(20,32,33,27)(22,29,34,43)(25,40,30,37)(26,38,36,46)(31,44,45,42)(39,48,41,47);(1,4,5,3)(2,6,8,7)(10,14)(13,16)(17,20,22,31,21,33,34,45)(18,25,26,41,23,30,36,39)(19,27,29,42,28,32,43,44)(24,37,38,48,35,40,46,47);(2,8)(6,7)(18,23)(19,28)(25,30)(26,36)(27,32)(29,43)(39,41)(42,44)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=f,d^2=fhi,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=dh,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, 7518003);