#8751488

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	C₂×#127	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4143 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=ghi,d^2=ghi,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,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, 8658684);