#503129

order	512
nilpotency class	4
isoclinism	isoclinic class of #501076
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 ◁ C₄×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1295 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 410325);