#512089

order	512
nilpotency class	4
isoclinism	isoclinic class of #511747
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	C₂×#466	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

982 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)(7,10)(8,9)(11,12)(14,19)(15,24)(17,29)(18,28)(20,34)(21,35)(22,38)(25,43)(26,44)(27,42)(30,31)(32,41)(36,39)(37,40);(6,9)(10,12)(13,14,16,19)(15,20,25,21)(17,22,27,32)(18,23,28,33)(24,35,43,34)(26,36,31,37)(29,41,42,38)(30,39,44,40);(2,4)(5,7,8,11)(6,10,9,12)(13,15,16,25)(14,20,19,21)(17,30,27,44)(18,31,28,26)(22,39,32,40)(23,37,33,36)(24,42,43,29)(34,41,35,38);(13,15,16,25)(14,21,19,20)(17,30,27,44)(18,31,28,26)(22,40,32,39)(23,36,33,37)(24,42,43,29)(34,38,35,41)

Polycyclic presentation:

2^9|b^2=i,c^2=e,d^2=i,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fh,g^a=gh,h^a=hi,d^b=di,f^c=fh,g^c=gh,h^c=hi,f^d=fh,g^d=gh,h^d=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, 419285);