#6701145

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1308 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)(11,15)(12,13)(14,16)(17,18,21,24)(19,27,30,23)(20,32,34,25)(22,26,36,40)(28,38,31,43)(29,39,47,45)(33,41,46,48)(35,44,37,42);(10,13)(11,16)(17,19)(18,23)(20,33)(21,30)(22,28)(24,27)(25,41)(26,43)(29,37)(31,36)(32,48)(34,46)(35,47)(38,40)(39,44)(42,45);(2,5)(6,8)(9,11,12,16)(10,14,13,15)(18,24)(19,28)(20,34)(22,36)(23,38)(27,43)(29,33)(30,31)(39,48)(41,45)(42,44)(46,47);(1,3,4,7)(2,6,5,8)(10,13)(14,15)(18,24)(20,34)(23,27)(26,40)(29,47)(33,46)(35,37)(38,43);(17,20,21,34)(18,25,24,32)(19,29,30,47)(22,37,36,35)(23,39,27,45)(26,44,40,42)(28,46,31,33)(38,41,43,48)

Polycyclic presentation:

2^9|a^2=i,c^2=fi,d^2=fg,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,e^c=ei,h^c=hi,e^d=ei,h^e=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6608341);