#8324949

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

4216 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:

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

Polycyclic presentation:

2^9|b^2=ghi,c^2=g,d^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8232145);