#7603174

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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1370 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)(17,18,23,28)(19,24,31,42)(20,32,37,47)(21,27,22,39)(25,38,41,30)(26,35,45,36)(29,44,34,43)(33,40,48,46);(2,5)(6,8)(9,11)(10,13)(12,16)(14,15)(17,19,23,31)(18,24,28,42)(20,33,37,48)(21,30,22,38)(25,39,41,27)(26,44,45,43)(29,35,34,36)(32,40,47,46);(10,14)(11,16)(17,19,21,38,23,31,22,30)(18,25,39,42,28,41,27,24)(20,34,35,33,37,29,36,48)(26,44,47,40,45,43,32,46);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(17,20,22,36,23,37,21,35)(18,26,27,32,28,45,39,47)(19,29,30,33,31,34,38,48)(24,40,41,44,42,46,25,43);(2,5)(6,8)(10,14)(13,15)(17,21,23,22)(18,27,28,39)(19,30,31,38)(20,35,37,36)(24,25,42,41)(26,32,45,47)(29,33,34,48)(40,43,46,44)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=hi,d^2=fg,e^2=i,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, 7510370);