#512093

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	C₂⁴	C₂×D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

961 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)(13,14,20,15)(16,29,32,43)(17,33,35,42)(18,36,39,44)(19,28,30,24)(21,37,25,31)(22,40,26,34)(23,41,27,38);(6,9)(10,12)(13,15,20,14)(16,25,32,21)(17,22,35,26)(18,27,39,23)(19,28,30,24)(29,31,43,37)(33,40,42,34)(36,38,44,41);(2,4)(5,7,8,11)(6,10,9,12)(13,16)(14,21)(15,25)(17,30)(18,34)(19,35)(20,32)(22,24)(23,42)(26,28)(27,33)(29,36)(31,38)(37,41)(39,40)(43,44);(13,17,20,35)(14,22,15,26)(16,30,32,19)(18,37,39,31)(21,24,25,28)(23,29,27,43)(33,44,42,36)(34,41,40,38)

Polycyclic presentation:

2^9|a^2=i,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, 419289);