#8463708

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

4321 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)(3,6)(4,7)(5,8)(9,10,15,18)(11,16,13,12)(14,17,23,22)(19,24,21,20)(25,26,31,30)(27,32,37,36)(28,38,33,29)(34,40,39,35);(3,8)(5,6)(9,11,15,13)(10,12,18,16)(14,19,23,21)(17,20,22,24)(25,27,31,37)(26,32,30,36)(28,34,33,39)(29,35,38,40);(1,3)(2,5)(4,8)(6,7)(9,12,15,16)(10,11,18,13)(14,20,23,24)(17,19,22,21)(25,28)(26,29)(27,34)(30,38)(31,33)(32,35)(36,40)(37,39);(1,3)(2,6)(4,8)(5,7)(9,10)(11,12)(13,16)(14,22)(15,18)(17,23)(19,24)(20,21)(25,27)(26,32)(28,34)(29,35)(30,36)(31,37)(33,39)(38,40);(9,13,15,11)(10,16,18,12)(14,21,23,19)(17,24,22,20)(25,29,31,38)(26,33,30,28)(27,35,37,40)(32,39,36,34);(1,2)(3,5)(4,7)(6,8)(9,14)(10,17)(11,19)(12,20)(13,21)(15,23)(16,24)(18,22)(25,30,31,26)(27,36,37,32)(28,38,33,29)(34,40,39,35)

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8370904);