#9057917

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

4407 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=ghi,d^2=h,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8965113);