#470218

order	512
nilpotency class	3
isoclinism	isoclinic class of #453996
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×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

763 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:

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

Polycyclic presentation:

2^9|a^2=eg,b^2=ei,c^2=efi,d^2=eg,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^c=de,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 377414);