#152219

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

Polycyclic presentation:

2^9|a^2=f,b^2=d,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, 59415);