#8375607

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4410 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,c^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8282803);