#8202312

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	#43531	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4277 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

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

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, 8109508);