#8657234

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

4599 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=hi,d^2=gi,e^2=gi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,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, 8564430);