#7534038

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

1355 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)(17,18,23,29)(19,25,34,42)(20,27,21,26)(22,28,24,30)(31,44,36,39)(32,48,47,40)(33,41,38,46)(35,45,37,43);(1,3,5,4)(2,6,8,7)(11,16)(13,15)(17,19)(18,25)(20,31)(21,36)(22,33)(23,34)(24,38)(26,39)(27,44)(28,41)(29,42)(30,46)(32,37)(35,47)(40,45)(43,48);(9,11,12,16)(10,13,14,15)(17,20,23,21)(18,26,29,27)(19,31,34,36)(22,37,24,35)(25,39,42,44)(28,45,30,43)(32,38,47,33)(40,46,48,41);(10,14)(13,15)(17,21,23,20)(18,27,29,26)(19,32,34,47)(22,37,24,35)(25,40,42,48)(28,45,30,43)(31,38,36,33)(39,46,44,41);(1,4,5,3)(2,7,8,6)(17,22,23,24)(18,28,29,30)(19,33,34,38)(20,35,21,37)(25,41,42,46)(26,43,27,45)(31,47,36,32)(39,48,44,40)

Polycyclic presentation:

2^9|a^2=fi,b^2=f,c^2=g,d^2=i,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, 7441234);