#146282

order	512
nilpotency class	4
isoclinism	isoclinic class of #146230
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2094
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#496
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

250 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 88 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:

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

Polycyclic presentation:

2^9|a^2=dgi,b^2=degh,c^2=g,d^2=g,e^2=gi,f^2=h,h^2=i,b^a=bd,c^a=ce,c^b=cf,d^b=dg,e^b=egi,f^b=fh,h^b=hi,d^c=dgi,e^c=egi,f^c=fhi,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, 53478);