#8680006

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	#52299	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4046 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=gh,d^2=i,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8587202);