#466451

order	512
nilpotency class	3
isoclinism	isoclinic class of #454426
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

900 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=efg,c^2=i,d^2=efi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^c=de,g^c=gi,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, 373647);