#9625451

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

4277 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=h,d^2=g,e^2=h,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, 9532647);