#10322705

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

4164 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,d^2=g,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi

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, 10229901);