#8139611

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

4277 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|b^2=g,c^2=h,d^2=gi,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,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, 8046807);