#8095858

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

4531 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)(11,16)(12,15)(13,17)(14,19)(18,23)(20,22)(21,24)(25,26,32,31)(27,29,36,33)(28,37,30,34)(35,40,38,39);(2,5)(6,7)(12,20)(15,22)(18,24)(21,23)(25,27)(26,33)(28,38)(29,31)(30,35)(32,36)(34,39)(37,40);(1,3,4,8)(2,6,5,7)(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23)(25,28,32,30)(26,34,31,37)(27,35,36,38)(29,39,33,40);(1,2)(3,6)(4,5)(7,8)(9,12)(10,15)(11,18)(13,20)(14,21)(16,23)(17,22)(19,24)(25,29)(26,27)(28,39)(30,40)(31,36)(32,33)(34,35)(37,38);(9,11)(10,16)(12,18)(13,19)(14,17)(15,23)(20,24)(21,22)(25,30,32,28)(26,34,31,37)(27,35,36,38)(29,40,33,39);(9,10)(11,14)(12,15)(13,17)(16,19)(18,21)(20,22)(23,24)(25,31,32,26)(27,29,36,33)(28,37,30,34)(35,39,38,40)

Polycyclic presentation:

2^9|a^2=g,c^2=g,e^2=i,f^2=i,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8003054);