#7610824

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1443 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)(19,24)(20,32)(21,26)(22,27)(23,28)(25,31)(29,40)(30,41)(33,39)(34,43)(35,42)(36,44)(37,45)(38,46)(47,48);(1,3,5,4)(2,6,8,7)(9,11)(10,13)(12,15)(14,16)(17,19,23,30)(18,24,28,41)(20,33,31,22)(21,35,37,34)(25,27,32,39)(26,42,45,43)(29,36,47,38)(40,44,48,46);(1,3,5,4)(2,7,8,6)(10,14)(11,15)(17,20)(18,25)(19,22)(21,35)(23,31)(24,39)(26,43)(27,41)(28,32)(29,36)(30,33)(34,37)(38,47)(40,46)(42,45)(44,48);(1,4,5,3)(2,6,8,7)(10,14)(13,16)(17,21,33,36,23,37,22,38)(18,26,39,44,28,45,27,46)(19,29,20,34,30,47,31,35)(24,40,32,43,41,48,25,42);(2,8)(6,7)(17,22,23,33)(18,27,28,39)(19,20,30,31)(21,36,37,38)(24,32,41,25)(26,44,45,46)(29,35,47,34)(40,42,48,43)

Polycyclic presentation:

2^9|b^2=fi,c^2=f,d^2=fhi,e^2=i,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,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, 7518020);