#8824442

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

4716 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,c^2=gi,d^2=gi,e^2=i,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8731638);