#8726682

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

4257 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=gh,d^2=gi,e^2=hi,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, 8633878);