#8095198

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

4319 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)(3,7)(4,6)(5,8)(9,10)(11,17)(12,16)(13,14)(15,19)(18,24)(20,23)(21,22)(25,26,30,29)(27,33,34,31)(28,32,36,37)(35,39,40,38);(1,2,4,6)(3,5,8,7)(9,11,15,12)(10,16,19,17)(13,20,18,22)(14,21,24,23)(27,34)(28,36)(31,33)(32,37);(1,2)(3,5)(4,6)(7,8)(9,11,15,12)(10,17,19,16)(13,22,18,20)(14,21,24,23)(25,27,30,34)(26,31,29,33)(28,35,36,40)(32,38,37,39);(1,3,4,8)(2,5,6,7)(9,12,15,11)(10,17,19,16)(13,22,18,20)(14,23,24,21)(25,28)(26,32)(27,35)(29,37)(30,36)(31,38)(33,39)(34,40);(9,13,15,18)(10,14,19,24)(11,20,12,22)(16,21,17,23)(25,26,30,29)(27,31,34,33)(28,32,36,37)(35,38,40,39);(9,14)(10,18)(11,21)(12,23)(13,19)(15,24)(16,22)(17,20)(25,29,30,26)(27,33,34,31)(28,37,36,32)(35,39,40,38)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=gi,d^2=g,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^c=eghi

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, 8002394);