#9224349

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

4156 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)(4,6)(5,9)(7,11)(10,16)(12,14)(13,15)(17,18,23,24)(19,26,20,25)(21,27,28,22)(29,30,31,32)(33,34)(35,40)(36,46)(37,38)(39,42)(41,45)(43,48)(44,47);(1,3)(2,7)(4,8)(5,10)(6,11)(9,15)(12,16)(13,14)(18,24)(21,28)(25,26)(29,31)(33,35)(34,40)(36,47)(37,43)(38,48)(39,45)(41,42)(44,46);(17,19,23,20)(18,25,24,26)(21,29,28,31)(22,30,27,32)(33,36,39,37)(34,38,42,46)(35,43,45,47)(40,44,41,48);(1,4,6,2)(3,8,11,7)(5,9,14,12)(10,15,13,16)(17,20,23,19)(18,26,24,25)(21,29,28,31)(22,30,27,32)(33,37,39,36)(34,38,42,46)(35,43,45,47)(40,48,41,44);(1,3)(2,8)(4,7)(5,13)(6,11)(9,15)(10,14)(12,16)(17,21)(18,27)(19,29)(20,31)(22,24)(23,28)(25,32)(26,30)(33,38)(34,37)(35,44)(36,42)(39,46)(40,47)(41,43)(45,48);(1,5,6,14)(2,9,4,12)(3,10,11,13)(7,15,8,16)(17,22,23,27)(18,28,24,21)(19,30,20,32)(25,31,26,29)(33,35)(34,41)(36,43)(37,47)(38,48)(39,45)(40,42)(44,46)

Polycyclic presentation:

2^9|a^2=hi,c^2=h,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, 9131545);