#10060943

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

4689 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=ghi,e^2=hi,f^2=gh,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, 9968139);