#506582

order	512
nilpotency class	4
isoclinism	isoclinic class of #500829
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×D₁₆	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄ ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

704 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 44 points:

(1,2,3,4)(5,6)(7,11)(8,9)(10,12)(13,14,17,23,32,41,31,21,19,25,16,22,18,24,33,40)(15,20,28,36,44,34,43,39,30,38,27,35,29,37,26,42);(6,9)(10,11)(13,15,17,28,32,44,31,43,19,30,16,27,18,29,33,26)(14,20,23,36,41,34,21,39,25,38,22,35,24,37,40,42);(2,4)(14,21)(15,26)(16,31)(17,33)(18,32)(20,34)(22,24)(23,41)(25,40)(27,44)(28,29)(30,43)(37,38)(39,42);(5,7,8,12)(6,10,9,11)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 413778);