#9054527

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

4376 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=g,d^2=h,e^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, 8961723);