#8807347

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	#50636	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4201 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:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=gh,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8714543);