#10058501

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

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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=i,d^2=g,e^2=ghi,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, 9965697);