#8180453

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

4573 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,d^2=g,e^2=gh,f^2=gh,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, 8087649);