#153231

order	512
nilpotency class	5
isoclinism	isoclinic class of #153230
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

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

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

normal subgroups only

This group has 351 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

32 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60427);