#8273857

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

4572 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)(11,15)(12,20)(13,14)(16,22)(17,19)(18,24)(21,23)(25,26)(27,35)(28,38)(29,40)(30,34)(31,37)(32,39)(33,36);(1,2)(3,6)(4,5)(7,8)(9,10)(11,17)(12,16)(13,14)(15,19)(18,24)(20,22)(21,23)(25,27)(26,31)(28,36)(29,39)(30,37)(32,33)(34,35)(38,40);(2,5)(6,8)(10,14)(15,17)(16,20)(23,24)(25,28,30,39)(26,32,34,38)(27,36,37,29)(31,33,35,40);(3,7)(6,8)(9,11,13,19)(10,15,14,17)(12,21,22,18)(16,24,20,23)(25,29)(26,33)(27,28)(30,36)(31,38)(32,35)(34,40)(37,39);(1,3)(2,6)(4,7)(5,8)(9,12,13,22)(10,16,14,20)(11,18,19,21)(15,23,17,24)(25,27)(26,31)(28,36)(29,39)(30,37)(32,33)(34,35)(38,40);(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,16)(13,14)(17,19)(18,23)(20,22)(21,24)(25,26)(27,31)(28,38)(29,33)(30,34)(32,39)(35,37)(36,40)

Polycyclic presentation:

2^9|c^2=g,d^2=gh,e^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,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, 8181053);