#8634816

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

4189 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)(3,8,13,16)(4,10,11,5)(6,12,15,14)(17,18,23,22)(19,24,26,21)(20,25,29,30)(27,31,32,28)(33,34)(35,45)(36,42)(37,39)(38,43)(40,44)(41,48)(46,47);(1,3,7,13)(2,8,9,16)(4,12,11,14)(5,6,10,15)(17,19)(18,24)(20,27)(21,22)(23,26)(25,31)(28,30)(29,32)(33,35,40,48)(34,41,44,45)(36,46,37,43)(38,39,47,42);(2,9)(3,13)(5,10)(12,14)(19,26)(21,24)(27,32)(28,31)(33,36,40,37)(34,42,44,39)(35,46,48,43)(38,45,47,41);(1,4,7,11)(2,10,9,5)(3,14,13,12)(6,16,15,8)(17,20)(18,25)(19,27)(21,28)(22,30)(23,29)(24,31)(26,32)(33,37,40,36)(34,42,44,39)(35,43,48,46)(38,45,47,41);(1,5,7,10)(2,11,9,4)(3,15,13,6)(8,12,16,14)(17,21)(18,19)(20,28)(22,26)(23,24)(25,27)(29,31)(30,32)(33,38,40,47)(34,43,44,46)(35,42,48,39)(36,41,37,45);(1,6,7,15)(2,12,9,14)(3,10,13,5)(4,8,11,16)(17,22,23,18)(19,21,26,24)(20,25,29,30)(27,31,32,28)(33,39,40,42)(34,37,44,36)(35,47,48,38)(41,43,45,46)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=g,d^2=gh,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=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, 8542012);