#8115588

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

4131 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=h,d^2=hi,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi,d^c=dgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8022784);