#7603185

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×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1363 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)(19,24)(20,31)(21,36)(22,38)(23,27)(25,35)(26,37)(28,45)(29,44)(30,40)(32,42)(33,39)(34,47)(41,46)(43,48);(2,5)(6,8)(9,11)(10,13)(12,16)(14,15)(17,19,23,30)(18,24,27,40)(20,32,34,33)(21,37,29,22)(25,41,43,45)(26,44,38,36)(28,35,46,48)(31,42,47,39);(10,14)(11,16)(17,20,22,35,23,34,37,48)(18,25,26,31,27,43,38,47)(19,28,29,33,30,46,21,32)(24,39,36,45,40,42,44,41);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(17,21,22,19,23,29,37,30)(18,24,26,36,27,40,38,44)(20,33,35,46,34,32,48,28)(25,42,31,41,43,39,47,45);(2,5)(6,8)(10,14)(13,15)(20,34)(25,43)(28,46)(31,47)(32,33)(35,48)(39,42)(41,45)

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,c^2=h,d^2=fg,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cgi,d^b=dh,g^b=gi,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, 7510381);