#8678858

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	#40129	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4363 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8586054);