#10521362

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

4207 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=g,d^2=i,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, 10428558);