#8103498

order	512
nilpotency class	2
isoclinism	isoclinic class of #53390
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂⁵
commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4552 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:

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

Polycyclic presentation:

2^9|b^2=gi,c^2=gi,d^2=g,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,d^c=dgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8010694);