#10322951

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

4365 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)(11,18)(12,22)(13,16)(14,23)(15,17)(19,24)(20,21)(25,26,31,35)(27,34,30,32)(28,29,33,40)(36,39,38,37);(1,2)(3,7)(4,5)(6,8)(9,11,15,13)(10,16,17,18)(12,20,23,19)(14,21,22,24)(25,26,31,35)(27,32,30,34)(28,29,33,40)(36,37,38,39);(2,5)(6,7)(9,12)(10,14)(11,19)(13,20)(15,23)(16,24)(17,22)(18,21)(25,27,31,30)(26,32,35,34)(28,36,33,38)(29,37,40,39);(2,5)(6,7)(9,12)(10,14)(11,20)(13,19)(15,23)(16,21)(17,22)(18,24)(25,28)(26,29)(27,36)(30,38)(31,33)(32,37)(34,39)(35,40);(1,2)(3,6)(4,5)(7,8)(9,13,15,11)(10,16,17,18)(12,20,23,19)(14,24,22,21)(25,29)(26,33)(27,37)(28,35)(30,39)(31,40)(32,38)(34,36);(1,3)(2,6)(4,8)(5,7)(9,14)(10,12)(11,21)(13,24)(15,22)(16,20)(17,23)(18,19)(25,30,31,27)(26,34,35,32)(28,38,33,36)(29,39,40,37)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10230147);