#10442522

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

4335 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:

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

Polycyclic presentation:

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

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, 10349718);