#486664

order	512
nilpotency class	4
isoclinism	isoclinic class of #486299
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	C₂²×C₈	C₂⁴
prime radical	C₂²×D₄	D₈
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

720 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 720 conjugacy classes of subgroups — too many to draw. Only its 138 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 393860);