#10088541

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

4427 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,d^2=g,e^2=g,f^2=gh,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, 9995737);