#8534662

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

4116 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)(3,11)(4,14)(5,10)(6,8)(7,13)(9,15)(12,16)(17,18,23,25)(19,24,21,20)(22,26,31,32)(27,30,29,28);(1,3,6,13)(2,7,8,11)(4,12,15,10)(5,9,16,14)(17,19,23,21)(18,24,25,20)(22,27,31,29)(26,30,32,28)(33,34)(35,42)(36,39)(37,40)(38,41)(43,47)(44,45)(46,48);(2,8)(4,15)(7,11)(10,12)(18,25)(19,21)(26,32)(27,29)(33,35,38,40)(34,37,41,42)(36,44,46,43)(39,47,48,45);(1,3)(2,7)(4,12)(5,14)(6,13)(8,11)(9,16)(10,15)(17,20,23,24)(18,19,25,21)(22,30,31,28)(26,29,32,27)(33,34)(35,37)(36,39)(38,41)(40,42)(43,45)(44,47)(46,48);(1,4)(2,9)(3,12)(5,11)(6,15)(7,16)(8,14)(10,13)(17,21,23,19)(18,24,25,20)(22,29,31,27)(26,30,32,28)(33,36,38,46)(34,39,41,48)(35,43,40,44)(37,45,42,47);(1,5,6,16)(2,10,8,12)(3,9,13,14)(4,11,15,7)(17,22)(18,26)(19,27)(20,28)(21,29)(23,31)(24,30)(25,32)(33,37)(34,40)(35,41)(36,45)(38,42)(39,44)(43,48)(46,47)

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=ghi,d^2=gh,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8441858);