#8943622

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

4512 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)(9,10,15,13)(11,17,12,16)(14,21,22,18)(19,23,20,24)(25,26,29,33)(27,32,34,30)(28,31,37,39)(35,40,36,38);(3,7)(6,8)(9,11,15,12)(10,16,13,17)(14,19,22,20)(18,23,21,24)(27,34)(30,32)(35,36)(38,40);(1,2)(3,6)(4,5)(7,8)(10,13)(14,22)(16,17)(19,20)(25,27,29,34)(26,30,33,32)(28,35,37,36)(31,38,39,40);(9,12,15,11)(10,17,13,16)(14,20,22,19)(18,24,21,23)(25,28)(26,31)(27,35)(29,37)(30,38)(32,40)(33,39)(34,36);(1,3)(2,6)(4,7)(5,8)(9,13,15,10)(11,16,12,17)(14,21,22,18)(19,23,20,24)(25,27)(26,32)(28,36)(29,34)(30,33)(31,38)(35,37)(39,40);(1,2)(3,6)(4,5)(7,8)(9,14)(10,18)(11,19)(12,20)(13,21)(15,22)(16,23)(17,24)(25,26)(27,32)(28,31)(29,33)(30,34)(35,40)(36,38)(37,39)

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=i,d^2=g,e^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, 8850818);