#10040416

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

4456 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=g,d^2=i,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,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, 9947612);