#8159397

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

4723 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)(3,5)(4,7)(6,8)(13,21)(16,22)(18,23)(19,24)(26,30)(27,33)(32,38)(34,35);(2,3)(6,7)(9,10)(11,12)(13,16)(14,15)(17,20)(18,19)(21,22)(23,24)(25,26)(27,31)(28,32)(29,30)(33,37)(34,39)(35,40)(36,38);(1,2,5,3)(4,6,8,7)(10,15)(11,17)(16,22)(18,23)(27,33)(31,37)(34,35)(39,40);(1,3,5,2)(4,7,8,6)(9,11,14,17)(10,12,15,20)(13,18,21,23)(16,19,22,24)(25,27)(26,31)(28,35)(29,33)(30,37)(32,40)(34,36)(38,39);(1,4,5,8)(2,6,3,7)(9,12)(10,11)(13,19)(14,20)(15,17)(16,18)(21,24)(22,23)(25,28,29,36)(26,32,30,38)(27,34,33,35)(31,39,37,40);(9,13)(10,16)(11,18)(12,19)(14,21)(15,22)(17,23)(20,24)(25,26)(27,31)(28,32)(29,30)(33,37)(34,39)(35,40)(36,38)

Polycyclic presentation:

2^9|c^2=h,d^2=i,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^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, 8066593);