#506289

order	512
nilpotency class	4
isoclinism	isoclinic class of #499897
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #376 ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

671 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 413485);