#8105571

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

4516 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=gh,d^2=g,f^2=ghi,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, 8012767);