#255029

order	512
nilpotency class	3
isoclinism	isoclinic class of #246503
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	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

952 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)(9,10)(11,20)(12,19)(13,18)(14,17)(15,22)(16,21)(23,24)(25,26,29,33)(27,35,38,31)(28,34,30,32)(36,40,37,39);(2,5)(6,8)(9,11)(10,15)(12,21)(13,22)(14,23)(16,17)(18,20)(19,24)(25,27)(26,31)(28,37)(29,38)(30,36)(32,39)(33,35)(34,40);(1,3,4,7)(2,6,5,8)(9,11)(10,16)(12,21)(13,22)(14,23)(15,19)(17,20)(18,24)(26,32)(27,36)(28,30)(33,34)(37,38)(39,40);(2,5)(6,8)(9,12,13,14)(10,17,18,19)(11,21,22,23)(15,16,20,24)(25,28,29,30)(26,32,33,34)(27,37,38,36)(31,39,35,40)

Polycyclic presentation:

2^9|a^2=i,c^2=eh,d^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,c^b=cg,f^b=fh,g^b=gi,d^c=di,f^c=fh,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 162225);