#8899115

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

4371 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=g,d^2=h,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8806311);