#7616125

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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1362 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)(4,8)(5,6)(9,10)(11,15)(12,14)(13,16)(17,18,21,25)(19,27,28,23)(20,31,34,24)(22,26,35,40)(29,36,33,39)(30,41,42,37)(32,45,46,38)(43,47,44,48);(2,6)(7,8)(9,11,12,16)(10,13,14,15)(19,28)(20,32)(22,35)(23,27)(24,38)(26,40)(29,43)(31,45)(33,44)(34,46)(36,47)(39,48);(1,3,5,4)(2,7,6,8)(10,14)(11,16)(17,19)(18,23)(20,33)(21,28)(22,30)(24,39)(25,27)(26,37)(29,34)(31,36)(32,43)(35,42)(38,47)(40,41)(44,46)(45,48);(1,4,5,3)(2,8,6,7)(10,14)(11,16)(17,20,22,32,21,34,35,46)(18,24,26,38,25,31,40,45)(19,29,30,44,28,33,42,43)(23,36,37,48,27,39,41,47);(2,6)(7,8)(10,14)(13,15)(19,28)(23,27)(29,33)(30,42)(36,39)(37,41)(43,44)(47,48)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=fgi,d^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=dg,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, 7523321);