#8193880

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

4429 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,d^2=gi,e^2=i,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

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, 8101076);