#8720818

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

4093 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)(3,4)(5,9)(6,7)(8,10)(11,12)(13,14)(15,16)(17,18)(19,21)(20,28)(22,25)(23,27)(24,29)(26,32)(30,31)(33,34,39,44)(35,42,37,40)(36,46,47,41)(38,43,48,45);(1,2)(3,10)(4,8)(5,9)(6,7)(11,14)(12,13)(15,16)(17,19)(18,23)(20,26)(21,25)(22,27)(24,32)(28,30)(29,31)(33,35,39,37)(34,40,44,42)(36,43,47,45)(38,41,48,46);(2,7)(3,8)(9,16)(11,13)(20,29)(24,28)(26,31)(30,32)(33,35,39,37)(34,40,44,42)(36,43,47,45)(38,41,48,46);(1,3,6,8)(2,4,7,10)(5,13,15,11)(9,14,16,12)(17,20,22,29)(18,24,25,28)(19,26,27,31)(21,30,23,32)(33,36,39,47)(34,41,44,46)(35,45,37,43)(38,42,48,40);(1,4)(2,8)(3,7)(5,14)(6,10)(9,11)(12,15)(13,16)(17,21,22,23)(18,19,25,27)(20,30,29,32)(24,26,28,31)(33,37,39,35)(34,42,44,40)(36,43,47,45)(38,41,48,46);(1,5,6,15)(2,9,7,16)(3,11,8,13)(4,12,10,14)(17,19)(18,21)(20,31)(22,27)(23,25)(24,32)(26,29)(28,30)(33,38,39,48)(34,43,44,45)(35,41,37,46)(36,40,47,42)

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=hi,d^2=gi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8628014);