#8658478

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4559 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,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, 8565674);