#8371372

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

4613 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)(11,12)(13,17)(14,15)(16,19)(18,20)(21,22)(23,24)(25,26,31,30)(27,32,34,28)(29,37,40,33)(35,36,38,39);(3,7)(5,8)(10,15)(12,16)(17,22)(20,23)(25,27)(26,32)(28,30)(29,38)(31,34)(33,36)(35,40)(37,39);(1,2)(3,5)(4,6)(7,8)(13,21)(17,22)(18,24)(20,23)(26,30)(27,34)(33,37)(35,38);(1,3)(2,5)(4,7)(6,8)(9,11,14,19)(10,16,15,12)(13,18,21,24)(17,23,22,20)(25,28)(26,27)(29,39)(30,34)(31,32)(33,38)(35,37)(36,40);(1,2)(3,5)(4,6)(7,8)(9,12,14,16)(10,11,15,19)(13,20,21,23)(17,18,22,24)(25,29)(26,33)(27,35)(28,36)(30,37)(31,40)(32,39)(34,38);(9,13,14,21)(10,17,15,22)(11,18,19,24)(12,20,16,23)(25,30,31,26)(27,28,34,32)(29,37,40,33)(35,36,38,39)

Polycyclic presentation:

2^9|a^2=h,d^2=g,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=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, 8278568);