#500677

order	512
nilpotency class	4
isoclinism	isoclinic class of #500676
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#459
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1205 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:

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

Polycyclic presentation:

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

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, 407873);