#503156

order	512
nilpotency class	4
isoclinism	isoclinic class of #501262
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

1154 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)(8,12)(9,15)(10,25)(11,27)(13,21)(14,19)(16,24)(18,34)(20,33)(22,31)(23,29)(26,32)(28,35)(30,36);(1,2,4,3)(6,12)(7,18)(9,19)(10,23)(11,24)(13,30)(14,29)(15,25)(17,32)(20,26)(21,22)(28,31)(33,34)(35,36);(5,7,19,18,11,22,23,35,8,20,10,26,24,36,9,21)(6,13,15,30,16,32,25,33,12,28,29,31,27,34,14,17);(1,3,4,2)(6,12)(13,28)(14,25)(15,29)(16,27)(17,33)(30,31)(32,34)

Polycyclic presentation:

2^9|b^2=ei,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=gh,h^a=hi,c^b=cg,f^b=fh,g^b=ghi,h^b=hi,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, 410352);