#8750834

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	#36677	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4240 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,d^2=hi,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8658030);