#8889380

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	#47651	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3943 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)(3,7,9,4)(5,13,15,10)(11,12,14,16)(18,20)(19,24)(22,30)(29,32)(33,34,39,43)(35,38,45,40)(36,46,44,41)(37,47,48,42);(1,3)(2,7)(4,8)(5,14)(6,9)(10,12)(11,15)(13,16)(17,18,23,20)(19,26,24,21)(22,25,30,28)(27,32,31,29)(33,35,39,45)(34,40,43,38)(36,37,44,48)(41,47,46,42);(2,8)(3,9)(10,13)(11,14)(17,19)(18,21)(20,26)(22,27)(23,24)(25,29)(28,32)(30,31)(33,36,39,44)(34,41,43,46)(35,37,45,48)(38,42,40,47);(1,3)(2,7)(4,8)(5,11)(6,9)(10,16)(12,13)(14,15)(17,20,23,18)(19,26,24,21)(22,28,30,25)(27,32,31,29)(33,37,39,48)(34,42,43,47)(35,44,45,36)(38,46,40,41);(1,4,6,7)(2,9,8,3)(5,12,15,16)(10,14,13,11)(17,21,23,26)(18,24,20,19)(22,29,30,32)(25,31,28,27)(33,35)(34,40)(36,37)(38,43)(39,45)(41,47)(42,46)(44,48);(1,5,6,15)(2,10,8,13)(3,11,9,14)(4,12,7,16)(17,22)(18,25)(19,27)(20,28)(21,29)(23,30)(24,31)(26,32)(33,38,39,40)(34,35,43,45)(36,42,44,47)(37,46,48,41)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8796576);