#8093178

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

4692 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8000374);