#8588885

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

4178 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=g,d^2=gh,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=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, 8496081);