#8096984

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

4538 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=g,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

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, 8004180);