#8723008

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

4598 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,d^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, 8630204);