#8185335

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

4476 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=gh,d^2=gi,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, 8092531);