#8107273

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
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4546 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)(3,7)(4,6)(5,8)(9,10,13,16)(11,17,19,14)(12,20,21,15)(18,22,24,23)(28,36)(32,37)(34,40)(38,39);(1,2,4,6)(3,5,8,7)(9,11,13,19)(10,14,16,17)(12,18,21,24)(15,22,20,23)(26,30)(31,33)(32,37)(38,39);(1,2)(3,5)(4,6)(7,8)(9,11,13,19)(10,14,16,17)(12,18,21,24)(15,22,20,23)(25,26)(27,33)(28,32)(29,30)(31,35)(34,39)(36,37)(38,40);(1,3)(2,5)(4,8)(6,7)(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,23)(21,24)(25,27,29,35)(26,31,30,33)(28,34,36,40)(32,38,37,39);(9,12,13,21)(10,15,16,20)(11,18,19,24)(14,22,17,23);(9,10)(11,14)(12,15)(13,16)(17,19)(18,22)(20,21)(23,24)(25,28)(26,32)(27,34)(29,36)(30,37)(31,38)(33,39)(35,40)

Polycyclic presentation:

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

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, 8014469);