#9029114

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

4132 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)(3,9)(4,12)(5,8)(6,11)(7,14)(10,15)(13,16)(17,18,23,26)(19,24,22,20)(21,25,31,32)(27,29,30,28)(34,36)(39,43)(40,44)(47,48);(1,2,5,8)(3,6,11,9)(4,7,14,12)(10,15,13,16)(17,19,23,22)(18,24,26,20)(21,27,31,30)(25,29,32,28)(33,34,38,36)(35,39,42,43)(37,40,46,44)(41,47,45,48);(1,3,5,11)(2,6,8,9)(4,13,14,10)(7,16,12,15)(17,20,23,24)(18,19,26,22)(21,29,31,28)(25,30,32,27)(33,35,38,42)(34,39,36,43)(37,45,46,41)(40,48,44,47);(1,3)(2,6)(4,10)(5,11)(7,15)(8,9)(12,16)(13,14)(17,20)(18,19)(21,29)(22,26)(23,24)(25,30)(27,32)(28,31)(33,36,38,34)(35,39,42,43)(37,44,46,40)(41,47,45,48);(1,2)(3,9)(4,7)(5,8)(6,11)(10,16)(12,14)(13,15)(17,21)(18,25)(19,27)(20,28)(22,30)(23,31)(24,29)(26,32)(33,35)(34,39)(36,43)(37,41)(38,42)(40,47)(44,48)(45,46);(1,4)(2,7)(3,10)(5,14)(6,15)(8,12)(9,16)(11,13)(17,22,23,19)(18,20,26,24)(21,30,31,27)(25,28,32,29)(33,37)(34,40)(35,41)(36,44)(38,46)(39,47)(42,45)(43,48)

Polycyclic presentation:

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

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, 8936310);