#8701695

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	#2416	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4277 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)(9,10,13,14)(11,15,18,22)(12,16,21,19)(17,23,20,24)(25,26)(27,33)(28,39)(29,34)(30,38)(31,36)(32,37)(35,40);(2,3)(6,7)(10,14)(15,22)(16,19)(23,24)(25,27,32,31)(26,33,37,36)(28,35,29,38)(30,39,40,34);(1,3,5,2)(4,7,8,6)(25,28,32,29)(26,34,37,39)(27,38,31,35)(30,33,40,36);(1,4)(2,6)(3,7)(5,8)(9,10)(11,15)(12,19)(13,14)(16,21)(17,24)(18,22)(20,23)(25,29,32,28)(26,34,37,39)(27,38,31,35)(30,36,40,33);(9,11,13,18)(10,15,14,22)(12,20,21,17)(16,24,19,23)(25,30,32,40)(26,35,37,38)(27,39,31,34)(28,33,29,36);(9,12,13,21)(10,16,14,19)(11,17,18,20)(15,23,22,24)(25,31,32,27)(26,36,37,33)(28,35,29,38)(30,39,40,34)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=i,d^2=h,e^2=gi,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, 8608891);