#8701431

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	#52297	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4347 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)(9,10,16,20)(11,21,15,17)(12,18,13,23)(14,19,24,22)(25,26,32,31)(27,34,28,33)(29,36,30,35)(37,39,38,40);(2,3)(6,7)(9,11,16,15)(10,17,20,21)(12,22,13,19)(14,18,24,23)(25,27,32,28)(26,33,31,34)(29,37,30,38)(35,39,36,40);(1,3,5,2)(4,7,8,6)(9,12,16,13)(10,18,20,23)(11,22,15,19)(14,17,24,21)(25,28,32,27)(26,34,31,33)(29,38,30,37)(35,40,36,39);(1,4)(2,6)(3,7)(5,8)(9,13,16,12)(10,18,20,23)(11,22,15,19)(14,21,24,17)(25,29,32,30)(26,35,31,36)(27,37,28,38)(33,39,34,40);(9,14)(10,19)(11,18)(12,17)(13,21)(15,23)(16,24)(20,22)(25,30,32,29)(26,36,31,35)(27,38,28,37)(33,40,34,39);(9,15,16,11)(10,17,20,21)(12,19,13,22)(14,18,24,23)(25,31,32,26)(27,34,28,33)(29,35,30,36)(37,39,38,40)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=g,d^2=gi,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,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, 8608627);