#489102

order	512
nilpotency class	4
isoclinism	isoclinic class of #12324
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#447
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂×#447
prime radical	#61251	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#436 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

689 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=i,e^2=gi,f^2=hi,g^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fhi,g^a=gi,e^b=eg,f^b=fh,g^b=gi,e^c=eh,f^c=fhi

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, 396298);