#10325721

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

4416 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=gh,e^2=h,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, 10232917);