#9713480

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

4114 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)(4,8)(6,7)(9,13)(10,12)(11,14)(15,16)(20,28)(24,31)(27,29)(30,32)(33,34,40,43)(35,44,37,38)(36,45,46,41)(39,47,48,42);(4,11)(8,14)(10,15)(12,16)(17,18)(19,26)(20,24)(21,23)(22,25)(27,32)(28,31)(29,30)(33,35,40,37)(34,38,43,44)(36,42,46,47)(39,45,48,41);(2,7)(3,9)(8,14)(10,15)(17,19,22,21)(18,23,25,26)(20,27,28,29)(24,30,31,32)(33,36,40,46)(34,41,43,45)(35,42,37,47)(38,39,44,48);(1,3,6,9)(2,5,7,13)(4,10,11,15)(8,12,14,16)(17,20)(18,24)(19,27)(21,29)(22,28)(23,30)(25,31)(26,32)(33,37,40,35)(34,38,43,44)(36,42,46,47)(39,41,48,45);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,18)(19,23)(20,24)(21,26)(22,25)(27,30)(28,31)(29,32)(33,38)(34,35)(36,39)(37,43)(40,44)(41,42)(45,47)(46,48);(1,5)(2,9)(3,7)(4,12)(6,13)(8,15)(10,14)(11,16)(17,21,22,19)(18,23,25,26)(20,29,28,27)(24,30,31,32)(33,39)(34,42)(35,45)(36,44)(37,41)(38,46)(40,48)(43,47)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gi,d^2=h,f^2=i,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, 9620676);