#7536311

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514040
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

1626 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)(20,25)(21,26)(22,28)(23,27)(29,38)(30,39)(31,41)(32,40)(33,42)(34,43)(35,44)(36,46)(37,45)(47,48);(2,5)(6,8)(9,11)(10,13)(12,16)(14,15)(17,19,23,32)(18,24,27,40)(20,29,21,34)(22,36,37,31)(25,38,26,43)(28,46,45,41)(30,33,47,35)(39,42,48,44);(1,3,4,7)(2,6,5,8)(10,14)(11,16)(17,20)(18,25)(19,29)(21,23)(22,33)(24,38)(26,27)(28,42)(30,31)(32,34)(35,37)(36,47)(39,41)(40,43)(44,45)(46,48);(10,14)(13,15)(17,21,23,20)(18,26,27,25)(19,30,32,47)(22,33,37,35)(24,39,40,48)(28,42,45,44)(29,36,34,31)(38,46,43,41);(2,5)(6,8)(10,14)(13,15)(18,27)(20,21)(24,40)(28,45)(29,34)(30,47)(33,35)(41,46)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=fg,d^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei,e^d=ei,h^d=hi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7443507);