#8185331

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

4460 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=gh,d^2=i,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

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, 8092527);