#152235

order	512
nilpotency class	5
isoclinism	isoclinic class of #149792
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2126
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	#532	C₂³
socle	C₂	#12797
prime radical	#3150	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ #2931 ◁ G
LECS	G ▷ #532 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

364 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 68 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)(3,15)(4,21,23,35)(5,26)(6,11)(7,13,30,40)(8,14)(9,17)(10,22,36,56)(12,29,38,27)(16,44,46,59)(18,34)(19,42,52,32)(20,43)(24,54,25,55)(28,39,57,37)(31,48,58,47)(33,51,60,49)(41,45,53,63)(50,62,64,61);(1,3,5,17,8,20,6,18)(2,9,11,15,14,34,26,43)(4,16,22,45,25,48,28,50)(7,19,27,49,30,52,29,51)(10,31,35,41,37,59,54,61)(12,32,13,33,38,42,40,60)(21,53,39,44,55,62,36,58)(23,46,56,63,24,47,57,64);(1,4)(2,10)(3,16)(5,22)(6,28)(7,23)(8,25)(9,31)(11,35)(12,21)(13,39)(14,37)(15,41)(17,45)(18,50)(19,46)(20,48)(24,30)(26,54)(27,56)(29,57)(32,53)(33,44)(34,59)(36,40)(38,55)(42,62)(43,61)(47,52)(49,63)(51,64)(58,60)

Polycyclic presentation:

2^9|a^2=f,b^2=d,d^2=i,e^2=h,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,g^a=gh,h^a=hi,e^c=eh,f^c=fghi,g^c=gi,f^e=fhi,g^f=gi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59431);