#9141669

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

4166 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=gi,d^2=h,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,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, 9048865);