#10225566

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

4362 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:

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

Polycyclic presentation:

2^9|b^2=hi,c^2=g,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,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, 10132762);