#8301323

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
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4304 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=gh,d^2=g,e^2=gi,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, 8208519);