#153170

order	512
nilpotency class	5
isoclinism	isoclinic class of #153169
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#13346
commutator, Frattini, cosocle	C₂×#124	C₂³
prime radical	#33568	C₂
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

549 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 549 conjugacy classes of subgroups — too many to draw. Only its 50 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:

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

Polycyclic presentation:

2^9|a^2=i,d^2=hi,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dhi,f^a=fghi,g^a=gi,c^b=cf,d^b=dhi,e^b=ei,f^b=fi,g^b=gh,d^c=dg,f^c=fi,g^c=gi,h^c=hi,f^d=fh,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, 60366);