#153304

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	#13424
prime radical	#3018	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

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

normal subgroups only

This group has 329 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=d,d^2=i,e^2=hi,f^2=i,g^2=i,b^a=bd,c^a=ce,e^a=ehi,f^a=fgh,g^a=gi,c^b=cf,d^b=di,e^b=eg,f^b=fi,g^b=gi,h^b=hi,e^c=ehi,g^c=gh,f^e=fh,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, 60500);