#9726523

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

4139 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)(3,5)(4,11)(6,14)(7,8)(9,16)(10,15)(12,13)(17,18)(19,25)(20,24)(21,26)(22,23)(27,32)(28,29)(30,31)(33,34)(35,41)(36,39)(37,40)(38,43)(42,47)(44,46)(45,48);(1,3,7,4)(2,5,8,11)(6,10,16,12)(9,13,14,15)(17,19)(18,21)(20,28)(22,26)(23,25)(24,32)(27,30)(29,31)(33,35)(34,38)(36,44)(37,43)(39,47)(40,41)(42,45)(46,48);(2,8)(3,4)(9,14)(10,12)(18,23)(19,26)(24,31)(27,28)(33,34,37,40)(35,38,43,41)(36,39,45,48)(42,46,44,47);(1,4,7,3)(2,5,8,11)(6,12,16,10)(9,13,14,15)(17,19)(18,21)(20,27)(22,26)(23,25)(24,29)(28,30)(31,32)(33,34)(35,38)(36,39)(37,40)(41,43)(42,46)(44,47)(45,48);(1,5)(2,3)(4,8)(6,15)(7,11)(9,12)(10,14)(13,16)(17,20)(18,24)(19,27)(21,29)(22,30)(23,31)(25,32)(26,28)(33,36,37,45)(34,39,40,48)(35,42,43,44)(38,46,41,47);(1,6,7,16)(2,9,8,14)(3,10,4,12)(5,13,11,15)(17,21)(18,19)(20,29)(22,25)(23,26)(24,27)(28,31)(30,32)(33,35)(34,38)(36,44)(37,43)(39,47)(40,41)(42,45)(46,48)

Polycyclic presentation:

2^9|b^2=gh,c^2=hi,d^2=gh,e^2=hi,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, 9633719);