#8700538

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

4439 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=gi,d^2=g,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,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, 8607734);