#8636361

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

4230 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=h,d^2=gh,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=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, 8543557);