#8122944

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

4266 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)(6,8)(9,10,14,18)(11,17,13,15)(12,16,20,23)(19,24,21,22)(25,26,31,32)(27,33,28,30)(29,37,39,34)(35,38,36,40);(1,2)(3,6)(4,5)(7,8)(9,11,14,13)(10,15,18,17)(12,19,20,21)(16,22,23,24)(26,32)(27,28)(34,37)(35,36);(2,5)(6,8)(12,20)(16,23)(19,21)(22,24)(25,27,31,28)(26,30,32,33)(29,35,39,36)(34,38,37,40);(9,10)(11,15)(12,16)(13,17)(14,18)(19,22)(20,23)(21,24)(25,28,31,27)(26,33,32,30)(29,36,39,35)(34,40,37,38);(1,2)(3,6)(4,5)(7,8)(9,12,14,20)(10,16,18,23)(11,19,13,21)(15,22,17,24)(25,29,31,39)(26,34,32,37)(27,35,28,36)(30,38,33,40);(1,3,4,7)(2,6,5,8)(9,13,14,11)(10,17,18,15)(12,21,20,19)(16,24,23,22)(25,30,31,33)(26,27,32,28)(29,38,39,40)(34,35,37,36)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=h,d^2=h,e^2=g,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=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, 8030140);