#153396

order	512
nilpotency class	5
isoclinism	isoclinic class of #153396
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#13516
commutator, Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂³	#510
LCS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #405 ◁ #2108 ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

157 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 157 conjugacy classes of subgroups — too many to draw. Only its 66 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) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

(1,2,9,30,23,42,56,32,5,12,28,41,29,43,55,25,8,14,34,21,26,22,27,40,24,4,6,11,7,13,33,54)(3,15,31,36,46,37,60,63,16,10,35,57,52,51,64,59,20,38,49,61,48,62,50,53,18,39,17,44,19,45,58,47);(1,3,5,16,8,20,24,18)(2,10,4,15,14,39,12,38)(6,17,9,31,28,35,34,49)(7,19,23,46,29,52,26,48)(11,36,21,44,41,61,30,57)(13,37,22,45,43,62,42,51)(25,53,32,59,54,63,40,47)(27,50,33,58,56,60,55,64);(1,4,5,14,8,12,24,2)(3,10,16,15,20,39,18,38)(6,25,9,32,28,54,34,40)(7,13,23,22,29,43,26,42)(11,27,21,33,41,56,30,55)(17,47,31,53,35,59,49,63)(19,51,46,37,52,45,48,62)(36,58,44,60,61,64,57,50)

Polycyclic presentation:

2^9|a^2=eh,b^2=d,c^2=d,d^2=i,e^2=ghi,f^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,c^b=cf,f^b=fi,e^c=eg,f^c=fi,g^c=gh,h^c=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320815, #72321401, #72321458
maximal normal subgroup of	#72320815, #72321401, #72321458

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60592);