#8935750

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	#57587	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4225 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=g,d^2=gh,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8842946);