#9712470

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

4761 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:

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

Polycyclic presentation:

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

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, 9619666);