#10521361

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

4372 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)(11,18)(12,16)(13,21)(14,15)(17,22)(19,23)(20,24)(25,26)(27,33)(28,34)(29,40)(30,35)(31,36)(32,37)(38,39);(10,15)(16,18)(17,21)(23,24)(25,27,32,30)(26,33,37,35)(28,36,38,29)(31,39,40,34);(1,2)(3,6)(4,5)(7,8)(9,11,14,12)(10,16,15,18)(13,19,22,20)(17,23,21,24)(25,28)(26,34)(27,29)(30,36)(31,35)(32,38)(33,40)(37,39);(3,7)(6,8)(9,10)(11,16)(12,18)(13,17)(14,15)(19,23)(20,24)(21,22)(25,29,32,36)(26,31,37,40)(27,38,30,28)(33,34,35,39);(1,3,4,7)(2,6,5,8)(9,12,14,11)(10,16,15,18)(13,19,22,20)(17,24,21,23)(25,30,32,27)(26,35,37,33)(28,36,38,29)(31,39,40,34);(1,2)(3,6)(4,5)(7,8)(9,13)(10,17)(11,19)(12,20)(14,22)(15,21)(16,23)(18,24)(25,31)(26,36)(27,39)(28,35)(29,37)(30,34)(32,40)(33,38)

Polycyclic presentation:

2^9|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, 10428557);