#153412

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

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

normal subgroups only

This group has 187 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) = 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);(1,3,5,4)(2,6,7,8)(9,11,12,15)(10,13,14,16)(17,18,22,27)(19,23,31,38)(20,32,34,48)(21,35,33,26)(24,39,41,28)(25,42,43,29)(30,44,45,37)(36,47,40,46);(1,4,5,3)(2,6,7,8)(10,14)(13,16)(18,23)(19,28)(20,29)(21,33)(24,25)(26,44)(27,38)(30,47)(31,39)(32,36)(34,42)(35,37)(40,48)(41,43)(45,46)

Polycyclic presentation:

2^9|a^2=d,b^2=dei,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, 60608);