#8505306

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

4637 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=gi,d^2=hi,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8412502);