#9720531

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) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=gi,e^2=g,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, 9627727);