#9214246

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

4151 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=gh,c^2=h,d^2=h,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^d=egi,f^d=fgi,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, 9121442);