#8460661

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

4459 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=gh,e^2=gi,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, 8367857);