#152193

order	512
nilpotency class	5
isoclinism	isoclinic class of #152191
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#12759
commutator	#127	C₂²×C₄
Frattini, cosocle	#532	C₂³
prime radical	#61111	C₂
LCS	G ▷ #127 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #532 ◁ G
LECS	G ▷ #532 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

801 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 801 conjugacy classes of subgroups — too many to draw. Only its 72 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=hi,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gh,h^a=hi,d^b=dhi,e^b=eghi,f^b=fgh,g^b=gi,d^c=dgi,g^c=gi,h^c=hi,e^d=ehi,f^d=fhi,g^e=gi,g^f=gi

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59389);