#8750790

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

4121 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=g,d^2=gi,e^2=h,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, 8657986);