#8722235

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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8629431);