#9730282

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

3941 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)(3,6)(4,7)(5,10)(8,13)(9,12)(11,15)(14,16)(17,18,22,20)(19,23,27,24)(21,28,29,25)(26,31,32,30)(33,34)(35,44)(36,42)(37,43)(38,41)(39,40)(45,47)(46,48);(1,2,6,3)(4,9,12,7)(5,8,13,10)(11,15,16,14)(17,19)(18,23)(20,24)(21,26)(22,27)(25,30)(28,31)(29,32)(34,40)(37,47)(41,44)(42,48);(2,3)(7,9)(8,10)(14,15)(17,19,22,27)(18,24,20,23)(21,26,29,32)(25,31,28,30)(33,35,39,38)(34,41,40,44)(36,43,46,45)(37,42,47,48);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,19)(18,24)(20,23)(21,26)(22,27)(25,31)(28,30)(29,32)(33,36,39,46)(34,42,40,48)(35,45,38,43)(37,44,47,41);(1,4,6,12)(2,7,3,9)(5,11,13,16)(8,14,10,15)(17,20,22,18)(19,23,27,24)(21,25,29,28)(26,31,32,30)(33,37)(34,43)(35,42)(36,44)(38,48)(39,47)(40,45)(41,46);(1,5,6,13)(2,8,3,10)(4,11,12,16)(7,14,9,15)(17,21,22,29)(18,25,20,28)(19,26,27,32)(23,30,24,31)(33,38,39,35)(34,44,40,41)(36,43,46,45)(37,42,47,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=gi,d^2=g,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=di,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, 9637478);