#9028671

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

4273 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=i,d^2=gh,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8935867);