#8103930

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

4489 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,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, 8011126);