#8186947

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

4456 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=h,d^2=gi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8094143);