#513135

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	#33733	C₂
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

317 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 317 conjugacy classes of subgroups — too many to draw. Only its 116 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)(18,23)(19,27)(21,32)(22,34)(24,38)(25,40)(26,31)(28,45)(29,41)(30,44)(33,36)(35,42)(37,39)(43,48)(46,47);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,18,20,23)(19,24,28,37)(21,25,30,35)(22,26,31,34)(27,39,45,38)(29,36,43,47)(32,42,44,40)(33,41,46,48);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(17,19,20,28)(18,24,23,37)(21,33,30,46)(22,36,31,47)(25,41,35,48)(26,43,34,29)(27,44,45,32)(38,42,39,40);(2,8)(6,9)(7,16)(14,15)(17,18,20,23)(19,24,28,37)(21,25,30,35)(22,26,31,34)(27,39,45,38)(29,36,43,47)(32,42,44,40)(33,41,46,48)

Polycyclic presentation:

2^9|b^2=e,c^2=e,d^2=i,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, 420331);