#8664338

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

4636 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:

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

Polycyclic presentation:

2^9|a^2=ghi,c^2=gh,d^2=gh,e^2=gh,f^2=gh,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, 8571534);