#8824121

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

4122 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)(3,9)(4,8)(5,6)(7,10)(11,15)(12,13)(14,16)(17,18,22,25)(19,21,28,31)(20,24,30,23)(26,32,27,29)(34,37)(39,42)(40,45)(46,48);(2,6)(7,9)(8,13)(14,15)(17,19,22,28)(18,21,25,31)(20,27,30,26)(23,32,24,29)(36,44)(40,45)(41,47)(46,48);(3,10)(4,12)(7,9)(8,13)(17,20,22,30)(18,23,25,24)(19,26,28,27)(21,29,31,32)(33,34,38,37)(35,39,43,42)(36,40,44,45)(41,46,47,48);(1,3)(2,7)(4,11)(5,10)(6,9)(8,14)(12,16)(13,15)(17,20)(18,24)(19,27)(21,29)(22,30)(23,25)(26,28)(31,32)(33,35,38,43)(34,39,37,42)(36,41,44,47)(40,46,45,48);(1,4,5,12)(2,8,6,13)(3,11,10,16)(7,14,9,15)(17,19)(18,21)(20,27)(22,28)(23,32)(24,29)(25,31)(26,30)(33,36,38,44)(34,40,37,45)(35,41,43,47)(39,46,42,48);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,21)(18,19)(20,29)(22,31)(23,26)(24,27)(25,28)(30,32)(33,37,38,34)(35,42,43,39)(36,45,44,40)(41,48,47,46)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=h,d^2=hi,e^2=ghi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8731317);