#296864

order	512
nilpotency class	3
isoclinism	isoclinic class of #271643
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

724 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 204060);