#10171644

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

4210 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10078840);