#10181188

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

4314 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=hi,c^2=gi,d^2=i,e^2=ghi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

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, 10088384);