#8649034

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
prime radical	#2414	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3941 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=ghi,d^2=hi,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, 8556230);