#8701470

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	#2541	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4355 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)(9,10,15,14)(11,18,12,16)(13,21,22,17)(19,23,20,24)(25,26,32,36)(27,37,31,33)(28,34,29,39)(30,35,40,38);(2,3)(6,7)(9,10,15,14)(11,18,12,16)(13,17,22,21)(19,24,20,23)(25,27,32,31)(26,33,36,37)(28,38,29,35)(30,34,40,39);(1,3,5,2)(4,7,8,6)(9,11,15,12)(10,16,14,18)(13,19,22,20)(17,23,21,24)(25,28,32,29)(26,34,36,39)(27,38,31,35)(30,33,40,37);(1,4)(2,6)(3,7)(5,8)(9,12,15,11)(10,16,14,18)(13,20,22,19)(17,23,21,24)(25,29,32,28)(26,34,36,39)(27,38,31,35)(30,37,40,33);(9,13,15,22)(10,17,14,21)(11,19,12,20)(16,23,18,24)(25,30)(26,35)(27,34)(28,33)(29,37)(31,39)(32,40)(36,38);(9,14,15,10)(11,18,12,16)(13,17,22,21)(19,23,20,24)(25,31,32,27)(26,33,36,37)(28,35,29,38)(30,34,40,39)

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=gh,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, 8608666);