#8750006

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	#42422	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4578 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=hi,d^2=gh,e^2=g,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, 8657202);