#8722297

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

4117 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=gh,d^2=g,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,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, 8629493);