#8094736

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

4259 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=g,d^2=h,e^2=i,f^2=g,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, 8001932);