#7616799

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 ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1304 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)(3,6)(4,5)(7,8)(9,10,12,14)(11,15,16,13)(17,18,21,25)(19,27,28,23)(20,24,33,39)(22,26,34,40)(29,38,32,35)(30,41,42,36)(31,37,45,48)(43,47,44,46);(2,5)(6,8)(9,11,12,16)(10,13,14,15)(19,28)(20,31)(22,34)(23,27)(24,37)(26,40)(29,43)(32,44)(33,45)(35,46)(38,47)(39,48);(1,3,4,7)(2,6,5,8)(10,14)(11,16)(17,19)(18,23)(20,32)(21,28)(22,30)(24,38)(25,27)(26,36)(29,33)(31,43)(34,42)(35,39)(37,46)(40,41)(44,45)(47,48);(10,14)(13,15)(17,20,22,31,21,33,34,45)(18,24,26,37,25,39,40,48)(19,29,30,44,28,32,42,43)(23,35,36,47,27,38,41,46);(2,5)(6,8)(10,14)(13,15)(19,28)(23,27)(29,32)(30,42)(35,38)(36,41)(43,44)(46,47)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=fgi,d^2=h,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, 7523995);