#444120

order	512
nilpotency class	3
isoclinism	isoclinic class of #418500
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	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

809 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=efi,c^2=eg,d^2=e,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,g^b=gi,h^d=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, 351316);