#8942812

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

4660 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)(16,22)(17,18)(19,24)(20,23)(25,26,30,34)(27,35,37,31)(28,32,38,39)(29,33,36,40);(1,2)(3,5)(4,6)(7,8)(9,11)(10,12)(13,19)(14,17)(15,18)(16,20)(21,24)(22,23)(25,27,30,37)(26,31,34,35)(28,36,38,29)(32,33,39,40);(1,2,4,6)(3,7,8,5)(10,15)(11,17)(13,22)(20,24)(28,38)(29,36)(32,39)(33,40);(3,8)(5,7)(9,12)(10,11)(13,20)(14,18)(15,17)(16,19)(21,23)(22,24)(25,26)(27,31)(28,39)(29,33)(30,34)(32,38)(35,37)(36,40);(1,3)(2,5)(4,8)(6,7)(9,10)(11,18)(12,17)(13,16)(14,15)(19,24)(20,23)(21,22)(25,28)(26,32)(27,36)(29,37)(30,38)(31,33)(34,39)(35,40);(1,2)(3,5)(4,6)(7,8)(9,13)(10,16)(11,19)(12,20)(14,22)(15,21)(17,23)(18,24)(25,29,30,36)(26,33,34,40)(27,28,37,38)(31,39,35,32)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,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, 8850008);