#8678981

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	#2537	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4299 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=i,d^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, 8586177);