#10143785

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

4425 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)(3,4)(5,8)(6,7)(9,10)(11,15)(12,20)(13,14)(16,22)(17,19)(18,24)(21,23)(25,26,32,30)(27,35,31,33)(28,34,29,38)(36,40,37,39);(1,3)(2,6)(4,8)(5,7)(9,10)(11,17)(12,16)(13,14)(15,19)(18,24)(20,22)(21,23)(25,27,32,31)(26,33,30,35)(28,36,29,37)(34,39,38,40);(1,4)(2,7)(3,8)(5,6)(10,14)(15,17)(16,20)(23,24)(25,28,32,29)(26,34,30,38)(27,36,31,37)(33,39,35,40);(2,8)(4,6)(9,10)(11,17)(12,16)(13,14)(15,19)(18,24)(20,22)(21,23)(25,29,32,28)(26,34,30,38)(27,37,31,36)(33,39,35,40);(1,4)(2,3)(5,6)(7,8)(9,11,13,19)(10,15,14,17)(12,21,22,18)(16,24,20,23)(25,30,32,26)(27,35,31,33)(28,38,29,34)(36,40,37,39);(1,3)(2,6)(4,8)(5,7)(9,12,13,22)(10,16,14,20)(11,18,19,21)(15,23,17,24)(25,31,32,27)(26,33,30,35)(28,37,29,36)(34,39,38,40)

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=ghi,d^2=ghi,e^2=gh,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, 10050981);