#9127667

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
prime radical	C₂×#120	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3986 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=gh,e^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9034863);