#8445145

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

4502 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8352341);