#8112407

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

4497 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8019603);