#8464263

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

4553 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)(3,5)(4,7)(6,8)(9,10)(11,15)(12,22)(13,18)(14,19)(16,20)(17,21)(23,24)(25,26)(27,32)(28,33)(29,39)(30,35)(31,36)(34,38)(37,40);(2,3)(6,7)(9,11,16,14)(10,15,20,19)(12,23,21,13)(17,18,22,24)(25,26)(27,37)(28,33)(29,34)(30,35)(31,36)(32,40)(38,39);(1,2,5,3)(4,6,8,7)(9,12)(10,17)(11,13)(14,23)(15,24)(16,21)(18,19)(20,22)(25,27,31,40)(26,32,36,37)(28,38,35,29)(30,39,33,34);(1,3,5,2)(4,7,8,6)(9,13,16,23)(10,18,20,24)(11,21,14,12)(15,17,19,22)(25,28)(26,33)(27,38)(29,40)(30,36)(31,35)(32,34)(37,39);(1,4,5,8)(2,6,3,7)(9,14,16,11)(10,19,20,15)(12,23,21,13)(17,18,22,24)(25,29,31,38)(26,34,36,39)(27,28,40,35)(30,37,33,32);(9,15,16,19)(10,11,20,14)(12,24,21,18)(13,22,23,17)(25,30,31,33)(26,35,36,28)(27,39,40,34)(29,37,38,32)

Polycyclic presentation:

2^9|b^2=gh,c^2=ghi,d^2=h,e^2=i,f^2=gi,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, 8371459);