#8993590

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

4276 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)(3,7)(4,5)(6,8)(9,10,14,13)(11,15,20,18)(12,16,21,22)(17,23,19,24)(26,30)(27,35)(28,38)(34,40);(3,8)(6,7)(9,11)(10,15)(12,19)(13,18)(14,20)(16,24)(17,21)(22,23)(25,26,31,30)(27,36,35,32)(28,33,38,39)(29,40,37,34);(1,3,5,8)(2,6,4,7)(9,12,14,21)(10,16,13,22)(11,17,20,19)(15,23,18,24)(25,27,31,35)(26,32,30,36)(28,37,38,29)(33,40,39,34);(3,8)(6,7)(9,13,14,10)(11,18,20,15)(12,16,21,22)(17,23,19,24)(25,28)(26,33)(27,37)(29,35)(30,39)(31,38)(32,40)(34,36);(1,3)(2,7)(4,6)(5,8)(9,12)(10,16)(11,19)(13,22)(14,21)(15,24)(17,20)(18,23)(25,29,31,37)(26,34,30,40)(27,28,35,38)(32,33,36,39);(1,4,5,2)(3,6,8,7)(9,11)(10,15)(12,19)(13,18)(14,20)(16,24)(17,21)(22,23)(25,30,31,26)(27,36,35,32)(28,39,38,33)(29,40,37,34)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=i,d^2=hi,e^2=g,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, 8900786);