#290138

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, prime radical	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

511 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=efh,b^2=efi,c^2=h,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, 197334);