#9712214

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

4033 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9619410);