#8471806

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

4661 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)(9,10,15,20)(11,17,12,16)(13,22,23,18)(14,19,24,21)(26,30)(27,33)(32,37)(34,39);(3,7)(5,8)(9,11,15,12)(10,16,20,17)(13,19,23,21)(14,18,24,22)(25,26,29,30)(27,31,33,36)(28,32,35,37)(34,38,39,40);(1,2)(3,5)(4,6)(7,8)(25,26)(27,31)(28,32)(29,30)(33,36)(34,38)(35,37)(39,40);(1,3,4,7)(2,5,6,8)(9,12,15,11)(10,17,20,16)(13,19,23,21)(14,18,24,22)(25,26)(27,31)(28,32)(29,30)(33,36)(34,38)(35,37)(39,40);(1,2)(3,5)(4,6)(7,8)(9,13)(10,18)(11,21)(12,19)(14,16)(15,23)(17,24)(20,22)(25,27)(26,31)(28,34)(29,33)(30,36)(32,38)(35,39)(37,40);(9,14,15,24)(10,19,20,21)(11,18,12,22)(13,16,23,17)(25,28,29,35)(26,32,30,37)(27,34,33,39)(31,38,36,40)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8379002);