#8436275

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

4462 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:

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

Polycyclic presentation:

2^9|b^2=gi,c^2=gi,d^2=gh,f^2=g,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, 8343471);