#506533

order	512
nilpotency class	4
isoclinism	isoclinic class of #500839
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	C₂×#466	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

787 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)(8,10)(9,12)(13,14,18,23,34,26,20,37,17,22,33,41,19,15,24,27)(16,28,31,25,36,29,44,39,30,43,38,40,32,42,35,21);(5,7,8,12)(6,9,10,11)(13,15,18,27,34,14,20,23,17,26,33,37,19,22,24,41)(16,29,31,39,36,43,44,40,30,42,38,21,32,28,35,25);(2,4)(13,16,17,30)(14,21,22,39)(15,25,26,40)(18,35,33,44)(19,36,34,32)(20,38,24,31)(23,42,41,29)(27,28,37,43);(6,10)(9,11)(14,22)(15,26)(21,39)(23,41)(25,40)(27,37)(28,43)(29,42)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 413729);