#8817747

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	#52004	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4750 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)(6,8)(9,10)(11,16)(12,15)(13,18)(14,17)(19,23)(20,21)(22,24)(26,28)(27,30)(33,36)(34,37);(3,7)(6,8)(11,18)(13,16)(19,20)(21,23)(25,26,31,28)(27,32,30,35)(29,36,38,33)(34,40,37,39);(1,2)(3,6)(4,5)(7,8)(9,11,14,18)(10,13,17,16)(12,20,22,19)(15,23,24,21)(25,27,31,30)(26,32,28,35)(29,37,38,34)(33,40,36,39);(9,12,14,22)(10,15,17,24)(11,19,18,20)(13,21,16,23)(25,28,31,26)(27,32,30,35)(29,36,38,33)(34,39,37,40);(1,3)(2,6)(4,7)(5,8)(9,11)(10,16)(12,19)(13,17)(14,18)(15,23)(20,22)(21,24)(25,29,31,38)(26,33,28,36)(27,34,30,37)(32,39,35,40);(1,2)(3,6)(4,5)(7,8)(9,13,14,16)(10,11,17,18)(12,21,22,23)(15,19,24,20)(25,30,31,27)(26,32,28,35)(29,37,38,34)(33,39,36,40)

Polycyclic presentation:

2^9|b^2=g,c^2=gi,d^2=gi,e^2=g,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, 8724943);