#489788

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
prime radical	#19283	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#436 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 689 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)(3,11)(4,10)(5,8)(6,9)(7,12)(13,16)(14,15)(17,18)(19,27)(20,31)(21,36)(22,24)(23,28)(25,35)(26,34)(29,43)(30,45)(32,42)(33,46)(37,41)(38,44)(39,47)(40,48);(1,3,6,14)(2,7,9,16)(4,13,5,12)(8,15,10,11)(17,19)(18,23)(20,32)(21,37)(22,29)(24,38)(25,41)(26,42)(27,31)(28,35)(30,34)(33,47)(36,39)(40,45)(43,46)(44,48);(2,8)(4,5)(7,15)(9,10)(11,16)(12,13)(20,33)(25,40)(27,43)(28,44)(31,46)(32,47)(35,48)(41,45);(2,9)(7,16)(8,10)(11,15)(18,24)(23,38)(25,40)(26,36)(27,43)(31,46)(39,42)(41,45)

Polycyclic presentation:

2^9|b^2=hi,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, 396984);