#8712350

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	C₂×#546	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4429 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=gh,d^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,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, 8619546);