#8254378

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	#39352	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4347 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)(11,16)(12,18)(13,21)(14,20)(15,17)(19,24)(22,23)(25,26,29,30)(27,33,35,31)(28,32,37,38)(34,40,36,39);(1,2,5,4)(3,7,8,6)(9,11,15,14)(10,16,17,20)(12,21,22,19)(13,23,24,18)(26,30)(31,33)(32,38)(39,40);(2,4)(6,7)(10,17)(11,14)(18,23)(19,21)(25,27,29,35)(26,31,30,33)(28,36,37,34)(32,40,38,39);(9,12)(10,18)(11,19)(13,20)(14,21)(15,22)(16,24)(17,23)(25,26)(27,31)(28,32)(29,30)(33,35)(34,39)(36,40)(37,38);(1,3,5,8)(2,6,4,7)(9,13)(10,19)(11,18)(12,20)(14,23)(15,24)(16,22)(17,21)(25,26)(27,31)(28,32)(29,30)(33,35)(34,39)(36,40)(37,38);(1,4,5,2)(3,7,8,6)(9,14,15,11)(10,20,17,16)(12,21,22,19)(13,23,24,18)(25,28,29,37)(26,32,30,38)(27,34,35,36)(31,39,33,40)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=g,e^2=ghi,f^2=gi,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, 8161574);