#9041142

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	#46372	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4443 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8948338);