#146217

order	512
nilpotency class	4
isoclinism	isoclinic class of #144123
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	#493
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

307 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 307 conjugacy classes of subgroups — too many to draw. Only its 108 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) = 512

A minimal faithful permutation representation — 3 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=di,b^2=de,c^2=i,d^2=g,e^2=g,f^2=hi,h^2=i,b^a=bd,c^a=ce,c^b=cf,d^b=dg,e^b=eg,f^b=fh,h^b=hi,d^c=dg,e^c=eg,f^c=fh,h^c=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53413);