#142380

order	512
nilpotency class	4
isoclinism	isoclinic class of #142242
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1999
commutator, Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂×#466	C₂²
LCS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#424 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

418 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 70 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 — 3 generators on 48 points:

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

Polycyclic presentation:

2^9|b^2=h,c^2=g,d^2=gi,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,f^a=fh,g^a=gi,c^b=cf,d^b=dg,g^b=gi,d^c=dh,f^c=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 49576);