#503167

order	512
nilpotency class	4
isoclinism	isoclinic class of #501248
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 ◁ #2931 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1144 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)(7,17)(8,11)(9,14)(10,15)(12,24)(13,18)(16,27)(19,22)(20,31)(21,32)(23,26)(25,33)(28,36)(29,35)(30,34);(1,2,4,3)(5,7,9,18)(6,13,14,17)(8,22,23,36)(10,20,25,30)(11,28,26,19)(12,29,27,21)(15,34,33,31)(16,35,24,32);(5,8,11,6,12,15,25,16,9,23,26,14,27,33,10,24)(7,13,30,36,21,32,28,34,18,17,20,22,29,35,19,31);(1,3,4,2)(6,14)(8,23)(13,17)(15,33)(16,24)(22,36)(31,34)(32,35)

Polycyclic presentation:

2^9|b^2=e,c^2=fh,d^2=ei,f^2=hi,g^2=h,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=ghi,h^b=hi,d^c=di

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, 410363);