#7586433

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 ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1227 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fi,d^2=g,e^2=gi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cg,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, 7493629);