#9066037

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

4200 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=h,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^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, 8973233);