#9224381

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

4256 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=gh,d^2=i,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^d=egi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9131577);