#8751668

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	#2414	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4153 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=gh,d^2=gh,e^2=ghi,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, 8658864);