#8533648

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

4322 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=hi,d^2=gh,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=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, 8440844);