#8658475

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	#50737	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4317 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=g,e^2=gh,f^2=g,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, 8565671);