#8127327

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

4544 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=hi,d^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^c=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, 8034523);