#350053

order	512
nilpotency class	3
isoclinism	isoclinic class of #346898
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#25605	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

872 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=i,c^2=e,d^2=i,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=dg,f^a=fh,g^a=ghi,c^b=ch,f^c=fh,g^c=gi,f^d=fi,g^d=ghi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 257249);