#8462071

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

4600 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)(3,6)(4,7)(5,8)(12,20)(17,21)(18,24)(22,23)(26,32)(27,35)(28,37)(34,39);(3,8)(5,6)(9,10,14,13)(11,16,19,15)(12,17,20,21)(18,23,24,22)(25,26,31,32)(27,30,35,40)(28,38,37,33)(29,34,36,39);(1,3)(2,5)(4,8)(6,7)(9,11)(10,15)(12,18)(13,16)(14,19)(17,22)(20,24)(21,23)(25,27,31,35)(26,30,32,40)(28,36,37,29)(33,34,38,39);(1,3)(2,6)(4,8)(5,7)(9,11)(10,16)(12,18)(13,15)(14,19)(17,23)(20,24)(21,22)(25,28)(26,33)(27,36)(29,35)(30,34)(31,37)(32,38)(39,40);(9,12,14,20)(10,17,13,21)(11,18,19,24)(15,22,16,23)(25,29,31,36)(26,34,32,39)(27,28,35,37)(30,38,40,33);(1,2)(3,5)(4,7)(6,8)(9,13,14,10)(11,16,19,15)(12,21,20,17)(18,23,24,22)(25,30)(26,35)(27,32)(28,39)(29,38)(31,40)(33,36)(34,37)

Polycyclic presentation:

2^9|b^2=hi,c^2=g,e^2=hi,f^2=ghi,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, 8369267);