#489104

order	512
nilpotency class	4
isoclinism	isoclinic class of #488912
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	C₂²×D₄	D₈
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#436 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

685 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 685 conjugacy classes of subgroups — too many to draw. Only its 146 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) = 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)(17,18,22,24)(19,27,29,45)(20,31,33,47)(21,36,34,26)(23,38,39,28)(25,42,41,35)(30,46,37,43)(32,44,48,40);(1,3,6,14)(2,7,10,16)(4,13,5,12)(8,15,9,11)(17,19)(18,23)(20,32)(21,37)(22,29)(24,39)(25,43)(26,44)(27,31)(28,35)(30,34)(33,48)(36,40)(38,42)(41,46)(45,47);(2,8)(4,5)(7,15)(9,10)(11,16)(12,13)(20,33)(25,41)(27,45)(28,38)(31,47)(32,48)(35,42)(43,46);(18,24)(23,39)(25,41)(26,36)(27,45)(31,47)(40,44)(43,46)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,e^2=gi,f^2=hi,g^2=i,b^a=be,c^a=cf,d^a=di,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

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 396300);