#7589072

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1247 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fi,c^2=fi,d^2=fg,e^2=fg,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, 7496268);