#153413

order	512
nilpotency class	5
isoclinism	isoclinic class of #153078
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×D₁₆
commutator, Frattini, cosocle	C₂²×C₁₆	C₂³
LCS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

179 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 179 conjugacy classes of subgroups — too many to draw. Only its 76 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:

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

Polycyclic presentation:

2^9|a^2=di,b^2=de,c^2=d,f^2=gh,g^2=hi,h^2=i,b^a=bd,c^a=ce,c^b=cf,f^b=fg,g^b=gh,h^b=hi,f^c=fg,g^c=gh,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, 60609);