#9065957

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

4109 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,e^d=egi,f^d=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, 8973153);