#8096129

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

4572 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=g,d^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

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, 8003325);