#9214736

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

4154 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=h,d^2=g,e^2=g,f^2=g,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, 9121932);