#9224353

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

4018 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)(3,8)(4,14)(5,6)(7,10)(9,13)(11,16)(12,15)(17,18)(19,22)(20,25)(21,31)(23,26)(24,29)(27,30)(28,32)(33,34,37,40)(35,41,43,38)(36,39,44,46)(42,47,48,45);(1,3)(2,8)(4,15)(5,11)(6,16)(7,13)(9,10)(12,14)(17,19)(18,24)(20,27)(21,28)(22,26)(23,29)(25,32)(30,31)(36,44)(39,46)(42,48)(45,47);(1,4,7,5)(2,6,10,14)(3,11,13,15)(8,12,9,16)(17,20,23,21)(18,25,26,31)(19,27,29,28)(22,30,24,32)(33,35,37,43)(34,38,40,41)(36,42,44,48)(39,45,46,47);(1,5,7,4)(2,6,10,14)(3,11,13,15)(8,16,9,12)(17,21,23,20)(18,25,26,31)(19,28,29,27)(22,30,24,32);(1,6)(2,5)(3,12)(4,10)(7,14)(8,15)(9,11)(13,16)(17,18)(19,24)(20,25)(21,31)(22,29)(23,26)(27,32)(28,30)(33,36)(34,39)(35,42)(37,44)(38,45)(40,46)(41,47)(43,48);(1,3)(2,9)(4,11)(5,15)(6,16)(7,13)(8,10)(12,14)(17,22,23,24)(18,19,26,29)(20,30,21,32)(25,27,31,28)(33,34)(35,38)(36,39)(37,40)(41,43)(42,45)(44,46)(47,48)

Polycyclic presentation:

2^9|a^2=hi,c^2=gh,d^2=gi,f^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, 9131549);