#287265

order	512
nilpotency class	3
isoclinism	isoclinic class of #281531
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
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

579 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)(9,10,15,20,13,18,14,19)(11,21,23,24,12,16,22,17)(25,26,28,32,29,33,36,31)(27,30,35,37,34,38,40,39);(2,5)(6,7)(9,11,14,22,13,12,15,23)(10,16,19,24,18,21,20,17)(25,27,28,35,29,34,36,40)(26,30,32,37,33,38,31,39);(9,12,13,11)(10,17,18,24)(14,22,15,23)(16,19,21,20)(26,31)(27,34)(28,36)(30,37)(32,33)(38,39);(1,3,4,8)(2,6,5,7)(10,18)(16,21)(17,24)(19,20)

Polycyclic presentation:

2^9|a^2=ef,b^2=f,c^2=hi,d^2=ehi,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gi,f^c=fh,g^c=ghi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 194461);