#489847

order	512
nilpotency class	4
isoclinism	isoclinic class of #489727
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
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#446 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

753 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 753 conjugacy classes of subgroups — too many to draw. Only its 134 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)(3,9)(4,12)(5,7)(6,11)(8,13)(10,16)(14,15)(17,18)(19,28)(20,25)(21,33)(22,37)(23,26)(24,30)(27,35)(29,46)(31,45)(32,42)(34,39)(36,44)(38,43)(40,48)(41,47);(4,13)(8,12)(9,16)(11,14)(17,19)(18,24)(20,29)(21,34)(22,38)(23,31)(25,39)(26,40)(27,43)(28,33)(30,35)(32,48)(36,47)(37,41)(42,46)(44,45);(1,3,5,10)(2,6,7,15)(4,14,13,11)(8,16,12,9)(17,20,23,22)(18,25,26,37)(19,29,31,38)(21,35,36,32)(24,39,40,41)(27,44,42,33)(28,43,45,46)(30,47,48,34);(2,7)(6,15)(8,12)(9,16)(18,26)(24,40)(25,37)(27,42)(28,45)(33,44)(39,41)(43,46)

Polycyclic presentation:

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

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 397043);