#153227

order	512
nilpotency class	5
isoclinism	isoclinic class of #153227
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#13375
commutator, Frattini, cosocle	C₂×#124	C₂³
prime radical	#3545	C₂²
LCS	G ▷ C₂×#124 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#124 ◁ G
LECS	G ▷ C₂×#124 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 391 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)(3,8)(4,18,20,9)(5,23)(6,12)(7,10)(11,26)(13,42,43,64)(14,40)(15,46)(16,30)(17,32)(19,33,47,51)(21,55,24,58)(22,41,54,35)(25,39)(27,37)(28,36)(29,44,59,49)(31,50)(34,56,38,57)(45,61)(48,60)(52,63,53,62);(1,3)(2,8)(4,19,20,47)(5,15)(6,14)(7,17)(9,33,18,51)(10,32)(11,31)(12,40)(13,21,43,24)(16,27)(22,52,54,53)(23,46)(25,48)(26,50)(28,45)(29,34,59,38)(30,37)(35,63,41,62)(36,61)(39,60)(42,58,64,55)(44,57,49,56);(1,4)(2,9)(3,13)(5,24)(6,22)(7,20)(8,29)(10,18)(11,38)(12,41)(14,44)(15,47)(16,49)(17,43)(19,48)(21,25)(23,58)(26,57)(27,54)(28,56)(30,42)(31,51)(32,59)(33,61)(34,36)(35,37)(39,55)(40,64)(45,53)(46,63)(50,52)(60,62)

Polycyclic presentation:

2^9|a^2=d,b^2=d,f^2=i,g^2=i,b^a=bd,c^a=ce,e^a=ei,f^a=fgh,h^a=hi,c^b=cf,e^b=eg,g^b=gi,h^b=hi,d^c=di,f^c=fi,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, 60423);