#512091

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	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

999 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)(13,14,16,21)(15,24,25,19)(17,20,28,36)(18,31,29,23)(22,38,37,30)(26,33,41,32)(27,35,42,40)(34,43,39,44);(6,9)(10,12)(13,15)(14,19)(16,25)(17,26)(18,27)(20,32)(21,24)(22,34)(23,35)(28,41)(29,42)(30,43)(31,40)(33,36)(37,39)(38,44);(2,4)(5,7,8,11)(6,10,9,12)(14,20)(17,30)(18,29)(19,32)(21,36)(22,23)(24,33)(26,43)(27,42)(28,38)(31,37)(34,35)(39,40)(41,44);(14,20)(15,25)(17,30)(18,29)(19,33)(21,36)(22,23)(24,32)(26,44)(28,38)(31,37)(34,40)(35,39)(41,43)

Polycyclic presentation:

2^9|a^2=i,c^2=ei,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, 419287);