#153322

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, 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

241 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 241 conjugacy classes of subgroups — too many to draw. Only its 88 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=d,c^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.

Relationships

maximal subgroup of	#72321065, #72322863, #72322878
maximal normal subgroup of	#72321065, #72322863, #72322878
maximal quotient of	#72321194, #72322053

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60518);