#8258637

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

4226 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)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(17,18)(19,21)(20,28)(22,23)(24,26)(25,29)(27,31)(30,32)(33,34,39,38)(35,42,36,40)(37,45,46,41)(43,47,44,48);(2,6)(7,9)(8,13)(14,15)(18,23)(19,26)(20,29)(30,31)(35,36)(37,46)(40,42)(41,45);(4,12)(8,13)(10,16)(14,15)(18,23)(21,24)(25,28)(30,31)(33,35,39,36)(34,40,38,42)(37,43,46,44)(41,47,45,48);(1,3,5,11)(2,7,6,9)(4,10,12,16)(8,14,13,15)(17,19)(18,24)(20,27)(21,23)(22,26)(25,31)(28,30)(29,32)(33,36,39,35)(34,40,38,42)(37,44,46,43)(41,47,45,48);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,20,22,29)(18,25,23,28)(19,27,26,32)(21,30,24,31)(33,37)(34,41)(35,43)(36,44)(38,45)(39,46)(40,47)(42,48);(1,4,5,12)(2,8,6,13)(3,10,11,16)(7,14,9,15)(17,21,22,24)(18,19,23,26)(20,30,29,31)(25,27,28,32)(33,38,39,34)(35,40,36,42)(37,45,46,41)(43,47,44,48)

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^2=gh,e^2=hi,f^2=gi,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, 8165833);