#8104717

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

4517 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:

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

Polycyclic presentation:

2^9|b^2=gi,d^2=i,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, 8011913);