#7620206

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1225 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)(11,16)(12,15)(13,14)(17,18,21,25)(19,27,28,23)(20,24,34,40)(22,26,33,39)(29,38,32,35)(30,41,42,36)(31,37,45,48)(43,47,44,46);(1,3,5,4)(2,6,8,7)(9,10)(11,16)(12,15)(13,14)(19,28)(20,31)(22,33)(23,27)(24,37)(26,39)(29,43)(32,44)(34,45)(35,46)(38,47)(40,48);(1,3,5,4)(2,7,8,6)(10,14)(15,16)(17,19)(18,23)(20,32)(21,28)(22,30)(24,38)(25,27)(26,36)(29,34)(31,43)(33,42)(35,40)(37,46)(39,41)(44,45)(47,48);(9,11,13,12)(10,15,14,16)(17,20,33,45,21,34,22,31)(18,24,39,48,25,40,26,37)(19,29,42,43,28,32,30,44)(23,35,41,46,27,38,36,47);(1,4,5,3)(2,7,8,6)(9,12,13,11)(10,15,14,16)(19,28)(23,27)(29,32)(30,42)(35,38)(36,41)(43,44)(46,47)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=f,d^2=hi,e^2=fgi,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, 7527402);