#9054497

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

4497 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gi,d^2=h,e^2=i,f^2=g,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, 8961693);