#8720845

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

4007 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=i,d^2=ghi,e^2=g,f^2=h,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, 8628041);