#10183164

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
prime radical	#59033	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4346 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=ghi,d^2=ghi,e^2=gi,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, 10090360);