#500722

order	512
nilpotency class	4
isoclinism	isoclinic class of #499754
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

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

1105 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)(7,17)(8,14)(9,13)(10,25)(11,27)(12,19)(15,24)(16,22)(18,30)(20,33)(21,31)(23,29)(26,32)(28,34)(35,36);(1,2,4,3)(5,7,8,19)(6,12,14,17)(9,23,10,18)(11,28,22,21)(13,30,25,29)(15,33,32,36)(16,34,27,31)(20,24,35,26);(6,13)(7,18)(9,24)(10,26)(11,22)(14,25)(15,16)(19,23)(20,21)(27,32)(28,35)(29,36)(30,33)(31,34);(1,3,4,2)(6,14)(12,17)(13,25)(15,32)(16,27)(29,30)(31,34)(33,36)

Polycyclic presentation:

2^9|b^2=e,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,f^c=fh,g^c=gh,h^c=hi

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, 407918);