#10121167

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

4492 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:

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

Polycyclic presentation:

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

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, 10028363);