#8912243

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

4623 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=gh,d^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8819439);