#505647

order	512
nilpotency class	4
isoclinism	isoclinic class of #499897
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×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 ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

522 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2)(3,7)(4,5)(6,8)(9,10,12,14)(11,13,15,16)(17,18,20,25)(19,28,29,23)(21,24,32,39)(22,35,33,27)(26,41,40,34)(30,42,45,36)(31,38,46,44)(37,47,43,48);(2,5)(6,7)(9,11)(10,13)(12,15)(14,16)(17,19)(18,23)(20,29)(21,30)(22,31)(24,36)(25,28)(26,37)(27,38)(32,45)(33,46)(34,47)(35,44)(39,42)(40,43)(41,48);(11,15)(13,16)(18,24)(21,34)(22,33)(23,36)(25,39)(26,27)(28,42)(30,47)(31,46)(32,41)(35,40)(37,38)(43,44)(45,48);(1,3,4,8)(2,6,5,7)(18,25)(19,29)(24,39)(26,40)(27,35)(30,45)(31,46)(47,48)

Polycyclic presentation:

2^9|a^2=gi,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, 412843);