#9716819

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

4003 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)(4,11)(5,7)(8,12)(9,15)(10,13)(14,16)(17,18)(19,24)(20,29)(21,26)(22,30)(23,25)(27,32)(28,31)(33,34)(35,44)(36,39)(37,42)(38,41)(40,43)(45,48)(46,47);(1,3,5,10)(2,6,7,13)(4,9,12,16)(8,14,11,15)(17,19,23,27)(18,24,25,32)(20,21,30,31)(22,28,29,26)(33,35,40,38)(34,41,43,44)(36,45,37,46)(39,47,42,48);(2,7)(6,13)(8,11)(14,15)(18,25)(19,27)(20,30)(26,28)(33,36,40,37)(34,39,43,42)(35,45,38,46)(41,47,44,48);(1,2)(3,6)(4,8)(5,7)(9,14)(10,13)(11,12)(15,16)(17,20)(18,22)(19,21)(23,30)(24,28)(25,29)(26,32)(27,31)(33,37,40,36)(34,42,43,39)(35,46,38,45)(41,48,44,47);(1,3)(2,6)(4,9)(5,10)(7,13)(8,14)(11,15)(12,16)(17,21)(18,26)(19,20)(22,32)(23,31)(24,29)(25,28)(27,30)(33,38,40,35)(34,41,43,44)(36,46,37,45)(39,47,42,48);(1,4,5,12)(2,8,7,11)(3,9,10,16)(6,14,13,15)(17,22)(18,20)(19,28)(21,24)(23,29)(25,30)(26,27)(31,32)(33,39,40,42)(34,36,43,37)(35,47,38,48)(41,45,44,46)

Polycyclic presentation:

2^9|b^2=gh,c^2=g,d^2=g,e^2=g,f^2=ghi,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, 9624015);