#8247584

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

4346 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=hi,d^2=ghi,e^2=h,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, 8154780);