#153381

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) = 44
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=ei,b^2=d,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, 60577);