#513139

order	512
nilpotency class	5
isoclinism	isoclinic class of #513092
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁴
prime radical	#2215	C₂²
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

315 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 315 conjugacy classes of subgroups — too many to draw. Only its 118 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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

Polycyclic presentation:

2^9|b^2=ei,c^2=ei,f^2=gh,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fg,g^a=gh,h^a=hi,c^b=cei,f^c=fg,g^c=gh,h^c=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, 420335);