#9668344

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

4394 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)(3,6)(4,7)(5,10)(8,13)(9,12)(11,15)(14,16)(17,18)(19,24)(20,26)(21,27)(22,32)(23,25)(28,29)(30,31)(33,34,38,36)(35,41,43,39)(37,44,46,40)(42,47,45,48);(1,2,6,3)(4,9,12,7)(5,8,13,10)(11,15,16,14)(17,19,23,22)(18,24,25,32)(20,28,27,30)(21,31,26,29)(34,36)(39,41)(40,44)(47,48);(2,3)(7,9)(8,10)(14,15)(18,25)(20,27)(24,32)(28,30)(33,35,38,43)(34,39,36,41)(37,42,46,45)(40,47,44,48);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,20,23,27)(18,26,25,21)(19,28,22,30)(24,29,32,31)(33,35)(34,39)(36,41)(37,45)(38,43)(40,48)(42,46)(44,47);(1,4)(2,7)(3,9)(5,11)(6,12)(8,14)(10,15)(13,16)(17,21,23,26)(18,27,25,20)(19,29,22,31)(24,28,32,30)(33,36,38,34)(35,39,43,41)(37,44,46,40)(42,47,45,48);(1,5)(2,8)(3,10)(4,11)(6,13)(7,14)(9,15)(12,16)(17,22,23,19)(18,24,25,32)(20,30,27,28)(21,31,26,29)(33,37)(34,40)(35,42)(36,44)(38,46)(39,47)(41,48)(43,45)

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=gi,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, 9575540);