#8751674

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4068 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=g,d^2=g,e^2=gi,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, 8658870);