#10060947

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

4777 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)(9,10,14,11)(12,15,19,17)(13,16,22,18)(20,23,21,24)(25,26,31,34)(27,32,30,40)(28,33,38,29)(35,36,37,39);(1,3,5,8)(2,6,7,4)(10,11)(15,17)(16,18)(23,24)(25,27,31,30)(26,32,34,40)(28,35,38,37)(29,39,33,36);(2,7)(3,8)(12,19)(15,17)(20,21)(23,24)(25,28,31,38)(26,29,34,33)(27,35,30,37)(32,39,40,36);(1,3)(2,4)(5,8)(6,7)(9,11,14,10)(12,17,19,15)(13,18,22,16)(20,24,21,23)(25,28)(26,33)(27,35)(29,34)(30,37)(31,38)(32,36)(39,40);(1,3)(2,6)(4,7)(5,8)(9,12)(10,15)(11,17)(13,21)(14,19)(16,24)(18,23)(20,22)(25,29,31,33)(26,28,34,38)(27,36,30,39)(32,37,40,35);(1,4,5,6)(2,8,7,3)(9,13,14,22)(10,16,11,18)(12,20,19,21)(15,23,17,24)(25,30,31,27)(26,32,34,40)(28,37,38,35)(29,39,33,36)

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=hi,d^2=g,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, 9968143);