#10225940

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

4230 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=g,d^2=gh,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, 10133136);