#8402245

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

4417 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,d^2=i,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8309441);