#8703496

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

4263 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=h,d^2=h,e^2=gh,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, 8610692);