#8963628

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

4050 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=h,d^2=gi,e^2=i,f^2=g,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, 8870824);