#463452

order	512
nilpotency class	3
isoclinism	isoclinic class of #456400
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

755 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)(9,10,12,14)(11,13,15,16)(17,18,21,23)(19,27,29,42)(20,31,34,25)(22,36,33,26)(24,38,37,30)(28,44,32,43)(35,41,45,47)(39,46,40,48);(1,3,6,4)(2,7,5,8)(9,11)(10,13)(12,15)(14,16)(18,23)(24,37)(25,31)(26,36)(27,42)(39,40)(41,47)(43,44);(1,4,6,3)(2,7,5,8)(11,15)(13,16)(17,19,22,30,21,29,33,38)(18,24,26,27,23,37,36,42)(20,32,35,48,34,28,45,46)(25,40,41,44,31,39,47,43);(1,5,6,2)(3,8,4,7)(17,20,33,45,21,34,22,35)(18,25,36,47,23,31,26,41)(19,28,38,48,29,32,30,46)(24,39,42,44,37,40,27,43)

Polycyclic presentation:

2^9|a^2=eg,b^2=ei,c^2=efi,d^2=ef,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,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, 370648);