#8726721

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

4366 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)(4,8)(5,6)(25,26,31,35)(27,32,37,39)(28,38,30,33)(29,34,40,36);(2,6)(3,8)(9,10,14,11)(12,15,20,17)(13,16,22,18)(19,23,21,24)(25,27,31,37)(26,32,35,39)(28,34,30,36)(29,38,40,33);(1,3)(2,4)(5,8)(6,7)(10,11)(15,17)(16,18)(23,24)(25,28,31,30)(26,33,35,38)(27,36,37,34)(29,32,40,39);(1,3)(2,4)(5,8)(6,7)(9,11,14,10)(12,17,20,15)(13,16,22,18)(19,23,21,24)(25,29,31,40)(26,34,35,36)(27,33,37,38)(28,39,30,32);(1,4)(2,3)(5,7)(6,8)(9,12,14,20)(10,15,11,17)(13,21,22,19)(16,24,18,23)(25,26)(27,32)(28,33)(29,34)(30,38)(31,35)(36,40)(37,39);(9,13,14,22)(10,16,11,18)(12,19,20,21)(15,23,17,24)(25,30,31,28)(26,33,35,38)(27,36,37,34)(29,39,40,32)

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=gh,d^2=gi,e^2=hi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8633917);