#8270510

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

4120 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=hi,d^2=ghi,e^2=gi,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, 8177706);