#506522

order	512
nilpotency class	4
isoclinism	isoclinic class of #500834
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

805 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)(13,14,18,25,27,41,32,42,20,23,17,24,19,26,36,43)(15,21,29,22,35,39,33,44,30,37,28,38,16,31,34,40);(5,7,8,12)(6,9,10,11)(13,15,17,28,27,35,36,34,20,30,18,29,19,16,32,33)(14,21,24,38,41,39,43,40,23,37,25,22,26,31,42,44);(2,4)(13,16,20,35)(14,22,23,38)(15,27,30,19)(17,29,18,28)(21,24,37,25)(26,44,41,40)(31,43,39,42)(32,33,36,34);(6,10)(9,11)(14,23)(21,37)(22,38)(24,25)(26,41)(31,39)(40,44)(42,43)

Polycyclic presentation:

2^9|a^2=fi,b^2=egi,c^2=i,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,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, 413718);