#8750123

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×#555	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4165 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,c^2=ghi,d^2=g,e^2=h,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, 8657319);