#152217

order	512
nilpotency class	5
isoclinism	isoclinic class of #152215
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

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

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

normal subgroups only

This group has 425 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=d,d^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,g^a=gh,h^a=hi,f^c=fgi,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, 59413);