#8260862

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
prime radical	#56210	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3918 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)(3,9)(4,6)(5,12)(7,11)(8,14)(10,15)(13,16)(17,18,24,23)(19,27,22,25)(20,30,26,21)(28,29,31,32)(34,36)(40,43)(41,42)(47,48);(1,3)(2,7)(4,9)(5,13)(6,11)(8,16)(10,14)(12,15)(17,19,24,22)(18,25,23,27)(20,31,26,28)(21,32,30,29)(33,34,39,36)(35,43,46,40)(37,41,38,42)(44,48,45,47);(1,2,6,4)(3,7,11,9)(5,8,14,12)(10,15,13,16)(17,20)(18,21)(19,28)(22,31)(23,30)(24,26)(25,29)(27,32)(33,35,39,46)(34,40,36,43)(37,44,38,45)(41,47,42,48);(1,4,6,2)(3,7,11,9)(5,12,14,8)(10,15,13,16)(17,21,24,30)(18,26,23,20)(19,29,22,32)(25,31,27,28)(33,36,39,34)(35,43,46,40)(37,42,38,41)(44,48,45,47);(1,5)(2,8)(3,10)(4,12)(6,14)(7,15)(9,16)(11,13)(17,22,24,19)(18,27,23,25)(20,31,26,28)(21,32,30,29)(33,37,39,38)(34,41,36,42)(35,44,46,45)(40,47,43,48);(1,3)(2,9)(4,7)(5,10)(6,11)(8,16)(12,15)(13,14)(17,23,24,18)(19,27,22,25)(20,30,26,21)(28,32,31,29)(33,38,39,37)(34,42,36,41)(35,45,46,44)(40,48,43,47)

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,c^2=hi,d^2=g,e^2=i,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, 8168058);