#8633495

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

4109 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)(17,18)(19,24)(20,26)(21,30)(22,28)(23,25)(27,29)(31,32)(33,34)(35,42)(36,44)(37,38)(39,40)(41,47)(43,46)(45,48);(1,2,6,3)(4,9,11,7)(5,8,13,10)(12,15,16,14)(17,19)(18,24)(20,29)(21,31)(22,25)(23,28)(26,27)(30,32)(33,35,39,38)(34,37,40,42)(36,45,47,43)(41,48,44,46);(4,11)(7,9)(12,16)(14,15)(18,25)(19,28)(21,29)(26,32)(34,40)(37,42)(41,44)(46,48);(1,3,6,2)(4,9,11,7)(5,8,13,10)(12,14,16,15)(17,20)(18,26)(19,21)(22,27)(23,31)(24,30)(25,32)(28,29)(33,36,39,47)(34,41,40,44)(35,43,38,45)(37,46,42,48);(1,4)(2,7)(3,9)(5,12)(6,11)(8,14)(10,15)(13,16)(17,21)(18,27)(19,20)(22,26)(23,29)(24,32)(25,30)(28,31)(33,37)(34,35)(36,46)(38,40)(39,42)(41,43)(44,45)(47,48);(1,5)(2,8)(3,10)(4,12)(6,13)(7,14)(9,15)(11,16)(17,22)(18,28)(19,25)(20,30)(21,26)(23,24)(27,31)(29,32)(33,38,39,35)(34,37,40,42)(36,43,47,45)(41,48,44,46)

Polycyclic presentation:

2^9|b^2=g,d^2=g,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=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, 8540691);