#503159

order	512
nilpotency class	4
isoclinism	isoclinic class of #501256
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#436 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1174 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|b^2=e,c^2=fh,d^2=ei,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=ghi,h^a=hi,c^b=cg,f^b=fhi,g^b=gh,h^b=hi,g^c=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 410355);