#153209

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×D₁₆
commutator, Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂²×D₁₆	C₂²
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

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

normal subgroups only

This group has 271 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) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=di,e^2=gh,g^2=hi,h^2=i,b^a=bd,c^a=ce,e^a=eg,g^a=gh,h^a=hi,c^b=cf,e^c=eg,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, 60405);