#8720627

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

4210 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=gi,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, 8627823);