#8726409

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	C₂×#548	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4270 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,c^2=g,d^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, 8633605);