#8800258

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	#51780	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4547 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:

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

Polycyclic presentation:

2^9|a^2=ghi,c^2=g,d^2=gi,e^2=gi,f^2=g,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, 8707454);