#491032

order	512
nilpotency class	4
isoclinism	isoclinic class of #491025
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#459
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	C₂³	#459
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×M₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

432 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 128 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ei,c^2=i,e^2=g,f^2=hi,h^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fh,h^a=hi,d^b=dgi,d^c=dg,f^c=fh,h^c=hi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 398228);