#9668339

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4210 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=i,d^2=ghi,e^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^c=egi,e^d=egi,f^d=fgi

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, 9575535);