#10502807

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

4758 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=g,e^2=hi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10410003);