#8161280

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	#54264	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3982 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)(3,9,4,7)(5,13,14,10)(11,15,12,16)(17,18,24,23)(19,28,22,25)(20,27,21,26)(29,31,30,32)(33,34,38,42)(35,41,37,39)(36,40,46,44)(43,48,45,47);(1,3,6,4)(2,7,8,9)(5,11,14,12)(10,15,13,16)(17,19,24,22)(18,25,23,28)(20,30,21,29)(26,32,27,31)(33,35,38,37)(34,39,42,41)(36,43,46,45)(40,47,44,48);(2,8)(7,9)(10,13)(15,16)(17,20,24,21)(18,26,23,27)(19,29,22,30)(25,31,28,32)(33,36)(34,40)(35,43)(37,45)(38,46)(39,47)(41,48)(42,44);(1,2)(3,9)(4,7)(5,10)(6,8)(11,16)(12,15)(13,14)(17,21,24,20)(18,27,23,26)(19,30,22,29)(25,32,28,31)(33,34)(35,39)(36,44)(37,41)(38,42)(40,46)(43,48)(45,47);(1,4,6,3)(2,9,8,7)(5,12,14,11)(10,16,13,15)(17,22,24,19)(18,28,23,25)(20,30,21,29)(26,32,27,31)(33,36)(34,40)(35,43)(37,45)(38,46)(39,47)(41,48)(42,44);(1,5,6,14)(2,10,8,13)(3,11,4,12)(7,15,9,16)(17,23,24,18)(19,28,22,25)(20,27,21,26)(29,32,30,31)(33,37,38,35)(34,41,42,39)(36,45,46,43)(40,48,44,47)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=i,d^2=i,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8068476);