#8994379

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

4412 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=hi,d^2=gi,e^2=gh,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, 8901575);