#8993556

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

4425 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^2=hi,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8900752);