#9625402

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

4258 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9532598);