#8601998

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4200 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)(12,17)(13,15)(14,18)(19,22)(20,24)(21,23)(25,26,31,35)(27,29,37,32)(28,38,36,33)(30,34,40,39);(2,5)(6,8)(9,11,15,18)(10,16,13,14)(12,20,23,19)(17,24,21,22)(25,27,31,37)(26,32,35,29)(28,30,36,40)(33,34,38,39);(1,2,4,5)(3,6,7,8)(10,13)(11,18)(17,21)(19,20)(25,28,31,36)(26,33,35,38)(27,30,37,40)(29,39,32,34);(3,7)(6,8)(9,12,15,23)(10,17,13,21)(11,19,18,20)(14,24,16,22)(25,27)(26,32)(28,30)(29,35)(31,37)(33,34)(36,40)(38,39);(1,2)(3,6)(4,5)(7,8)(9,13,15,10)(11,16,18,14)(12,21,23,17)(19,22,20,24)(25,29,31,32)(26,27,35,37)(28,34,36,39)(30,33,40,38);(1,3)(2,6)(4,7)(5,8)(9,14,15,16)(10,18,13,11)(12,22,23,24)(17,19,21,20)(25,30)(26,34)(27,36)(28,37)(29,33)(31,40)(32,38)(35,39)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=hi,d^2=gh,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=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, 8509194);