#513130

order	512
nilpotency class	5
isoclinism	isoclinic class of #513086
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	C₂³	C₂×D₁₆
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

314 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 126 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)(17,18,20,26)(19,29,23,24)(21,25,32,42)(22,37,33,40)(27,45,43,34)(28,31,44,35)(30,38,36,41)(39,48,46,47);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,19,20,23)(18,24,26,29)(21,30,32,36)(22,31,33,35)(25,41,42,38)(27,39,43,46)(28,40,44,37)(34,48,45,47);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(18,25)(19,23)(21,34)(22,35)(24,38)(26,42)(27,28)(29,41)(30,47)(31,33)(32,45)(36,48)(37,39)(40,46)(43,44);(2,8)(6,9)(7,16)(14,15)(18,26)(24,29)(25,42)(27,43)(28,44)(37,40)(38,41)(39,46)

Polycyclic presentation:

2^9|a^2=i,b^2=e,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=ce,f^c=fg,g^c=gh,h^c=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420326);