#8750170

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

4485 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=i,d^2=h,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8657366);