#503981

order	512
nilpotency class	4
isoclinism	isoclinic class of #503866
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#481
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×D₄	SD₁₆
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#436 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

579 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)(5,6,10,14)(7,11,16,26)(8,19,22,12)(9,13,24,32)(15,29,20,25)(17,35,34,27)(18,28,23,31)(21,36,33,30);(1,2,4,3)(5,7,9,23,10,16,24,18)(6,11,13,31,14,26,32,28)(8,20,21,17,22,15,33,34)(12,29,30,27,19,25,36,35);(1,2,4,3)(7,15,16,20)(8,21)(9,24)(11,25,26,29)(12,30)(13,32)(17,18,34,23)(19,36)(22,33)(27,28,35,31);(1,3,4,2)

Polycyclic presentation:

2^9|a^2=i,b^2=fh,c^2=f,d^2=e,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,c^b=cg,f^b=fi,g^b=gh,h^b=hi,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, 411177);