#8096186

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

4552 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:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=g,e^2=h,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, 8003382);