#8754913

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

4329 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)(9,10,14,15)(11,13,18,16)(12,20,22,17)(19,23,24,21)(25,26,31,36)(27,32,39,38)(28,35,30,33)(29,34,37,40);(3,7)(5,8)(10,15)(13,16)(17,20)(21,23)(25,27)(26,32)(28,29)(30,37)(31,39)(33,40)(34,35)(36,38);(1,2)(3,5)(4,6)(7,8)(10,15)(11,18)(17,20)(19,24)(25,28,31,30)(26,33,36,35)(27,37,39,29)(32,34,38,40);(9,11)(10,16)(12,19)(13,15)(14,18)(17,23)(20,21)(22,24)(25,29,31,37)(26,34,36,40)(27,28,39,30)(32,35,38,33);(1,3,4,7)(2,5,6,8)(9,12,14,22)(10,17,15,20)(11,19,18,24)(13,21,16,23)(25,30,31,28)(26,35,36,33)(27,37,39,29)(32,34,38,40);(1,2)(3,5)(4,6)(7,8)(9,13,14,16)(10,11,15,18)(12,21,22,23)(17,19,20,24)(25,26)(27,38)(28,33)(29,34)(30,35)(31,36)(32,39)(37,40)

Polycyclic presentation:

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

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, 8662109);