#8114402

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

4236 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)(11,20)(12,13)(14,16)(15,17)(18,19)(21,24)(22,23)(26,30)(27,33)(32,37)(34,39);(1,2,4,6)(3,5,8,7)(9,11,15,14)(10,16,17,20)(12,23,19,21)(13,24,18,22)(25,26,29,30)(27,31,33,36)(28,32,35,37)(34,38,39,40);(1,2)(3,5)(4,6)(7,8)(10,17)(12,19)(16,20)(21,23)(25,26)(27,31)(28,32)(29,30)(33,36)(34,38)(35,37)(39,40);(1,3)(2,5)(4,8)(6,7)(9,12,15,19)(10,18,17,13)(11,21,14,23)(16,24,20,22)(25,26)(27,31)(28,32)(29,30)(33,36)(34,38)(35,37)(39,40);(9,13,15,18)(10,19,17,12)(11,22,14,24)(16,23,20,21)(25,27)(26,31)(28,34)(29,33)(30,36)(32,38)(35,39)(37,40);(9,14,15,11)(10,20,17,16)(12,23,19,21)(13,24,18,22)(25,28,29,35)(26,32,30,37)(27,34,33,39)(31,38,36,40)

Polycyclic presentation:

2^9|b^2=g,d^2=gi,e^2=gi,f^2=h,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, 8021598);