#9726249

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

3975 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)(3,8)(4,9)(5,13)(6,7)(10,15)(11,14)(12,16)(17,18)(19,24)(20,23)(21,25)(22,26)(27,30)(28,31)(29,32)(36,44)(40,47)(43,45)(46,48);(2,7)(4,12)(8,14)(10,13)(17,19,23,26)(18,24,20,22)(21,29,31,27)(25,32,28,30)(33,34)(35,42)(36,40)(37,39)(38,41)(43,48)(44,47)(45,46);(2,7)(3,11)(9,16)(10,13)(18,20)(19,26)(25,28)(27,29)(33,35,38,37)(34,39,41,42)(36,43,44,45)(40,46,47,48);(1,3,6,11)(2,8,7,14)(4,10,12,13)(5,9,15,16)(17,20,23,18)(19,22,26,24)(21,28,31,25)(27,32,29,30)(33,36)(34,40)(35,43)(37,45)(38,44)(39,46)(41,47)(42,48);(1,4)(2,9)(3,10)(5,14)(6,12)(7,16)(8,15)(11,13)(17,21,23,31)(18,25,20,28)(19,27,26,29)(22,32,24,30)(33,37,38,35)(34,39,41,42)(36,45,44,43)(40,46,47,48);(1,5,6,15)(2,10,7,13)(3,9,11,16)(4,8,12,14)(17,22)(18,19)(20,26)(21,30)(23,24)(25,29)(27,28)(31,32)(33,34)(35,39)(36,40)(37,42)(38,41)(43,46)(44,47)(45,48)

Polycyclic presentation:

2^9|b^2=gh,c^2=ghi,d^2=h,e^2=i,f^2=g,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, 9633445);