#153234

order	512
nilpotency class	5
isoclinism	isoclinic class of #153131
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2199
commutator, Frattini, cosocle	#532	C₂³
socle	C₂	#13377
prime radical	#3529	C₂²
LCS	G ▷ #532 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #532 ◁ G
LECS	G ▷ #532 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

360 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 46 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:

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

Polycyclic presentation:

2^9|a^2=dh,b^2=dh,d^2=i,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,f^a=fghi,h^a=hi,c^b=cf,d^b=di,e^b=eg,g^b=gi,h^b=hi,e^c=ei,g^c=gh,f^e=fhi,g^e=gi,g^f=gi

Conjugacy classes & character table

35 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60430);