#8664651

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

4521 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gi,d^2=h,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, 8571847);