#10325441

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

4527 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=g,e^2=h,f^2=g,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, 10232637);