#9214232

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

4156 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=h,d^2=h,e^2=h,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, 9121428);