#10219575

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

4380 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=h,d^2=gh,e^2=h,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, 10126771);