#9206399

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

4089 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)(3,5)(4,7)(6,12)(8,15)(9,10)(11,14)(13,16)(17,18)(19,28)(20,25)(21,26)(22,24)(23,27)(29,30)(31,32)(33,34,38,41)(35,42,37,39)(36,40,45,48)(43,47,44,46);(1,3)(2,5)(4,10)(6,11)(7,9)(8,13)(12,14)(15,16)(17,19,23,22)(18,24,27,28)(20,29,26,31)(21,30,25,32)(33,34)(35,39)(36,40)(37,42)(38,41)(43,46)(44,47)(45,48);(2,4)(3,9)(8,12)(11,16)(17,20,23,26)(18,25,27,21)(19,29,22,31)(24,32,28,30)(33,35,38,37)(34,39,41,42)(36,44,45,43)(40,47,48,46);(1,4,7,2)(3,10,9,5)(6,12,15,8)(11,14,16,13)(17,20)(18,25)(19,29)(21,27)(22,31)(23,26)(24,32)(28,30)(33,36,38,45)(34,40,41,48)(35,43,37,44)(39,46,42,47);(1,5,7,10)(2,3,4,9)(6,14,15,13)(8,16,12,11)(17,21,23,25)(18,26,27,20)(19,30,22,32)(24,31,28,29)(33,37,38,35)(34,39,41,42)(36,43,45,44)(40,47,48,46);(1,6,7,15)(2,8,4,12)(3,11,9,16)(5,13,10,14)(17,22,23,19)(18,24,27,28)(20,31,26,29)(21,30,25,32)(33,34)(35,39)(36,40)(37,42)(38,41)(43,46)(44,47)(45,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=g,d^2=h,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,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, 9113595);