#501991

order	512
nilpotency class	4
isoclinism	isoclinic class of #501810
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

535 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)(7,16)(10,21)(14,31)(17,33)(18,26)(20,24)(22,36)(32,35);(1,2,4,3)(5,6,8,13,9,15,11,25)(7,17,26,35,16,33,18,32)(10,22,20,31,21,36,24,14)(12,27,29,19,30,23,28,34);(5,7,9,16)(6,12,15,30)(8,18,11,26)(10,23,21,27)(13,28,25,29)(14,32,31,35)(17,22,33,36)(19,24,34,20);(1,3,4,2)(5,8,9,11)(6,13,15,25)(7,18,16,26)(10,20,21,24)(12,28,30,29)(14,22,31,36)(17,32,33,35)(19,27,34,23)

Polycyclic presentation:

2^9|b^2=efh,c^2=i,d^2=ei,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=ghi,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, 409187);