#9668377

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

4183 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)(3,7)(4,12)(5,6)(8,13)(9,10)(11,15)(14,16)(17,18,23,21)(19,20,27,24)(22,29,31,25)(26,30,32,28)(33,34)(35,42)(36,41)(37,38)(39,43)(40,47)(44,46)(45,48);(1,2)(3,9)(4,8)(5,6)(7,10)(11,15)(12,13)(14,16)(17,19,23,27)(18,24,21,20)(22,26,31,32)(25,30,29,28)(33,35)(34,40)(36,44)(37,45)(38,46)(39,47)(41,48)(42,43);(2,6)(3,10)(8,12)(11,16)(18,21)(20,24)(25,29)(28,30)(33,36,39,37)(34,41,43,38)(35,44,47,45)(40,48,42,46);(1,2)(3,9)(4,12)(5,6)(7,10)(8,13)(11,14)(15,16)(17,20)(18,19)(21,27)(22,30)(23,24)(25,32)(26,29)(28,31)(33,37,39,36)(34,41,43,38)(35,45,47,44)(40,48,42,46);(1,3)(2,7)(4,11)(5,10)(6,9)(8,14)(12,15)(13,16)(17,21,23,18)(19,24,27,20)(22,29,31,25)(26,30,32,28)(33,35)(34,42)(36,45)(37,44)(38,46)(39,47)(40,43)(41,48);(1,4,5,13)(2,8,6,12)(3,11,10,16)(7,14,9,15)(17,22)(18,25)(19,26)(20,28)(21,29)(23,31)(24,30)(27,32)(33,38,39,41)(34,37,43,36)(35,46,47,48)(40,45,42,44)

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=ghi,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, 9575573);