#10312587

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

4540 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=gh,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dgi,e^b=eghi,d^c=dghi,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, 10219783);