#8504795

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

4266 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=hi,d^2=gi,e^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8411991);