#8900133

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4482 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=g,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^c=fgi,e^d=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, 8807329);