#8731052

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	#35714	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4550 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,d^2=gh,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, 8638248);