#8757403

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
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4052 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:

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

Polycyclic presentation:

2^9|a^2=ghi,c^2=hi,d^2=g,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8664599);