#9779473

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	#2688	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4101 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=gh,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9686669);