#470865

order	512
nilpotency class	3
isoclinism	isoclinic class of #455107
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

674 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:

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

Polycyclic presentation:

2^9|a^2=eg,b^2=hi,c^2=ef,d^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^c=de,e^c=ei,f^c=fi,g^c=gi,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, 378061);