#505680

order	512
nilpotency class	4
isoclinism	isoclinic class of #500109
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	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

558 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)(5,6,11,14,9,15,10,24)(7,16,25,34,17,33,19,31)(8,20,26,30,21,36,23,13)(12,27,28,18,29,22,32,35);(1,2,4,3)(7,17)(8,21)(13,30)(16,33)(19,25)(20,36)(23,26)(31,34);(5,7,9,17)(6,12,15,29)(8,22,21,27)(10,25,11,19)(13,31,30,34)(14,32,24,28)(16,20,33,36)(18,23,35,26);(1,3,4,2)

Polycyclic presentation:

2^9|a^2=gh,b^2=e,c^2=i,d^2=e,f^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,h^a=hi,c^b=cg,f^b=fi,f^c=fhi,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, 412876);