#7535230

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514020
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1357 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=f,c^2=g,d^2=f,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cgi,d^b=dh,h^b=hi,e^c=ei,h^d=hi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7442426);