#8104725

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

4509 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)(12,17)(13,16)(14,18)(15,22)(20,23)(21,24)(26,31)(27,35)(28,29)(39,40);(1,2,4,5)(3,6,7,8)(9,11,14,22)(10,15,18,19)(12,20,13,21)(16,23,17,24)(25,26,30,31)(27,32,35,38)(28,34,29,33)(36,40,37,39);(2,5)(3,7)(9,10)(11,15)(12,17)(13,16)(14,18)(19,22)(20,24)(21,23)(25,27)(26,32)(28,37)(29,36)(30,35)(31,38)(33,40)(34,39);(1,3,4,7)(2,6,5,8)(9,12,14,13)(10,16,18,17)(11,20,22,21)(15,23,19,24)(25,28,30,29)(26,33,31,34)(27,36,35,37)(32,39,38,40);(2,5)(6,8)(9,12,14,13)(10,17,18,16)(11,21,22,20)(15,23,19,24)(25,29,30,28)(26,34,31,33)(27,36,35,37)(32,39,38,40);(9,13,14,12)(10,16,18,17)(11,21,22,20)(15,23,19,24)

Polycyclic presentation:

2^9|b^2=gi,d^2=gi,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, 8011921);