#8534943

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

4771 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=ghi,e^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8442139);