#8934442

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	#59037	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4410 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,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, 8841638);