#152220

order	512
nilpotency class	5
isoclinism	isoclinic class of #152219
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#12793
commutator	C₂×#41	C₂²×C₄
Frattini, cosocle	#532	C₂³
prime radical	#3018	C₂²
LCS	G ▷ C₂×#41 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #2931 ◁ G
LECS	G ▷ #532 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 427 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)(3,14)(4,20,23,55)(5,26)(6,12,29,38)(7,13)(8,17)(9,22,35,57)(10,15)(11,27,40,28)(16,45,48,30)(18,31,53,61)(19,43)(21,36,56,37)(24,34,25,39)(32,52,42,51)(33,44)(41,47,60,64)(46,54,63,62)(49,59,50,58);(1,3,15,44,7,19,5,17)(2,8,26,43,13,33,10,14)(4,16,22,47,25,50,21,46)(6,18,28,52,29,53,27,51)(9,30,55,62,37,59,34,41)(11,31,12,32,40,61,38,42)(20,54,36,58,39,60,35,45)(23,48,57,64,24,49,56,63);(1,4)(2,9)(3,16)(5,21)(6,23)(7,25)(8,30)(10,34)(11,39)(12,35)(13,37)(14,41)(15,22)(17,46)(18,48)(19,50)(20,40)(24,29)(26,55)(27,56)(28,57)(31,60)(32,45)(33,59)(36,38)(42,58)(43,62)(44,47)(49,53)(51,63)(52,64)(54,61)

Polycyclic presentation:

2^9|a^2=f,b^2=di,d^2=i,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,g^a=gh,h^a=hi,e^c=ei,f^c=fg,g^c=gi,h^c=hi,f^e=fhi,g^e=gi,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, 59416);