#8201724

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	#36725	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4822 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^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, 8108920);