#8105024

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

4544 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:

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

Polycyclic presentation:

2^9|b^2=gi,c^2=gi,d^2=gi,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, 8012220);