#8174312

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

4642 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=i,d^2=h,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi

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, 8081508);