#8904642

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	C₂³×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4297 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)(3,5,7,8)(9,10)(11,15)(12,20)(13,14)(16,21)(17,19)(18,23)(22,24)(26,31)(27,35)(28,38)(34,39);(3,7)(5,8)(9,10)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24)(25,26,30,31)(27,36,35,32)(28,33,38,40)(29,39,37,34);(1,2)(3,5)(4,6)(7,8)(10,14)(12,21)(15,17)(18,24)(25,27,30,35)(26,32,31,36)(28,37,38,29)(33,39,40,34);(9,10)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24)(25,28)(26,33)(27,37)(29,35)(30,38)(31,40)(32,39)(34,36);(1,3)(2,5)(4,7)(6,8)(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,23)(21,24)(25,29,30,37)(26,34,31,39)(27,28,35,38)(32,33,36,40);(1,2)(3,5)(4,6)(7,8)(9,12)(10,16)(11,18)(13,21)(14,20)(15,22)(17,23)(19,24)(25,28)(26,33)(27,37)(29,35)(30,38)(31,40)(32,39)(34,36)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=g,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8811838);