#144096

order	512
nilpotency class	4
isoclinism	isoclinic class of #144077
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2011
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#447
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

398 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 80 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)(3,9)(4,12)(5,8)(6,10)(7,14)(11,16)(13,15)(17,18,22,27)(19,28,31,46)(20,23,35,39)(21,36,34,26)(24,40,42,29)(25,43,38,32)(30,47,37,44)(33,45,48,41);(1,3,5,11)(2,6,8,15)(4,13,14,10)(7,16,12,9)(18,23)(20,32)(21,34)(24,28)(25,26)(27,39)(29,33)(30,37)(35,43)(36,38)(40,48)(41,47)(42,46)(44,45);(4,14)(6,15)(7,12)(10,13)(17,19,22,31)(18,24,27,42)(20,33,35,48)(21,37,34,30)(23,28,39,46)(25,44,38,47)(26,45,36,41)(29,43,40,32)

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=i,d^2=gi,e^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fghi,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=eh,g^b=gi,h^b=hi,d^c=dg,e^c=eh,g^c=gi,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 51292);