#8807307

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	#48390	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4351 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:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=g,d^2=g,e^2=gh,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, 8714503);