#8726395

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	#42617	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4442 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,c^2=g,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8633591);