#9723171

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

4107 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=gi,d^2=g,e^2=gh,f^2=gh,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, 9630367);