#8963023

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

4043 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=h,d^2=hi,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,f^c=fgi,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, 8870219);