#9001111

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

4419 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=g,d^2=g,e^2=i,f^2=h,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, 8908307);