#8260863

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

4074 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)(8,11)(10,12)(13,14)(15,16)(18,22)(19,25)(24,29)(26,31)(33,34,38,43)(35,42,45,39)(36,41,37,40)(44,47,46,48);(2,7)(4,8)(9,14)(12,15)(17,18,21,22)(19,23,25,28)(20,24,27,29)(26,30,31,32)(33,35)(34,39)(36,44)(37,46)(38,45)(40,47)(41,48)(42,43);(5,13)(9,14)(10,16)(12,15)(17,18)(19,23)(20,24)(21,22)(25,28)(26,30)(27,29)(31,32)(33,36,38,37)(34,40,43,41)(35,44,45,46)(39,47,42,48);(1,3)(2,8)(4,7)(5,10)(6,11)(9,15)(12,14)(13,16)(17,18)(19,23)(20,24)(21,22)(25,28)(26,30)(27,29)(31,32)(33,34)(35,42)(36,40)(37,41)(38,43)(39,45)(44,48)(46,47);(1,4,6,8)(2,3,7,11)(5,12,13,15)(9,10,14,16)(17,19)(18,23)(20,26)(21,25)(22,28)(24,30)(27,31)(29,32)(33,37,38,36)(34,41,43,40)(35,44,45,46)(39,47,42,48);(1,5)(2,9)(3,10)(4,12)(6,13)(7,14)(8,15)(11,16)(17,20,21,27)(18,24,22,29)(19,26,25,31)(23,30,28,32)(33,35)(34,42)(36,46)(37,44)(38,45)(39,43)(40,47)(41,48)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8168059);