#8104703

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

4528 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)(12,16)(13,18)(14,17)(15,21)(20,22)(23,24)(25,26)(27,33)(28,36)(29,32)(30,35)(31,37)(34,38)(39,40);(1,2,4,5)(3,6,7,8)(9,11)(10,15)(12,20)(13,22)(14,21)(16,23)(17,19)(18,24)(25,27)(26,30)(28,34)(29,35)(31,38)(32,33)(36,39)(37,40);(2,5)(3,7)(9,12,14,18)(10,16,17,13)(11,20,21,24)(15,23,19,22)(25,28,29,37)(26,31,32,36)(27,34,35,40)(30,38,33,39);(1,3,4,7)(2,6,5,8)(10,17)(12,18)(15,19)(20,24)(26,32)(27,35)(28,37)(38,39);(2,5)(6,8)(11,21)(15,19)(20,24)(22,23)(26,32)(27,35)(31,36)(34,40);(9,13,14,16)(10,18,17,12)(11,22,21,23)(15,24,19,20)

Polycyclic presentation:

2^9|b^2=i,c^2=g,d^2=i,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, 8011899);