#10143523

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

4633 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=i,d^2=ghi,e^2=h,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, 10050719);