#9625467

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

4247 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)(25,26,31,32)(27,29,34,30)(28,36,39,33)(35,38,40,37);(1,2)(3,4)(5,6)(7,8)(9,10,13,16)(11,14,19,17)(12,15,22,20)(18,23,21,24)(26,32)(27,34)(33,36)(35,40);(2,6)(3,7)(9,10,13,16)(11,17,19,14)(12,15,22,20)(18,24,21,23)(25,27)(26,29)(28,35)(30,32)(31,34)(33,37)(36,38)(39,40);(3,7)(4,8)(9,11,13,19)(10,14,16,17)(12,21,22,18)(15,24,20,23)(25,28,31,39)(26,33,32,36)(27,35,34,40)(29,37,30,38);(1,3)(2,4)(5,7)(6,8)(9,10)(11,17)(12,15)(13,16)(14,19)(18,24)(20,22)(21,23)(25,29,31,30)(26,34,32,27)(28,37,39,38)(33,40,36,35);(1,4)(2,3)(5,8)(6,7)(9,12,13,22)(10,15,16,20)(11,18,19,21)(14,23,17,24)(25,30,31,29)(26,27,32,34)(28,38,39,37)(33,35,36,40)

Polycyclic presentation:

2^9|a^2=gh,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, 9532663);