#501623

order	512
nilpotency class	4
isoclinism	isoclinic class of #501014
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#459
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#2213	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

693 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=e,d^2=ei,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gh,h^b=hi,f^c=fi,g^c=gh,h^c=hi

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, 408819);