#8772573

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

4372 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=ghi,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8679769);