#10090427

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

4401 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)(11,19)(12,21)(13,18)(14,22)(15,17)(16,20)(23,24)(25,26,32,35)(27,36,39,33)(28,30,29,40)(31,34,37,38);(1,2)(3,6)(4,7)(5,8)(9,11,15,20)(10,16,17,19)(12,18,22,24)(13,21,23,14)(25,27)(26,33)(28,31)(29,37)(30,38)(32,39)(34,40)(35,36);(1,3)(2,5)(4,8)(6,7)(10,17)(12,22)(16,19)(18,24)(25,28,32,29)(26,30,35,40)(27,31,39,37)(33,38,36,34);(1,2)(3,5)(4,7)(6,8)(9,12)(10,14)(11,18)(13,16)(15,22)(17,21)(19,23)(20,24)(25,29,32,28)(26,30,35,40)(27,37,39,31)(33,38,36,34);(1,3)(2,6)(4,8)(5,7)(9,13,15,23)(10,18,17,24)(11,14,20,21)(12,19,22,16)(25,30)(26,29)(27,38)(28,35)(31,36)(32,40)(33,37)(34,39);(1,2)(3,5)(4,7)(6,8)(9,14)(10,12)(11,13)(15,21)(16,18)(17,22)(19,24)(20,23)(25,31,32,37)(26,34,35,38)(27,28,39,29)(30,33,40,36)

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=gh,d^2=gh,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9997623);