#152611

order	512
nilpotency class	5
isoclinism	isoclinic class of #152610
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#13049
commutator	#124	C₂²×C₄
Frattini, cosocle	C₂×#124	C₂³
prime radical	#3153	C₂²
LCS	G ▷ #124 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#124 ◁ G
LECS	G ▷ C₂×#124 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

432 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 60 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)(3,8)(4,17,20,52)(5,22,24,37)(6,11,26,39)(7,12)(9,19,33,53)(10,25)(13,40,43,63)(14,45)(15,48,49,29)(16,30)(18,51,21,36)(23,28,58,57)(27,42,61,54)(31,46,60,64)(32,55,34,56)(35,47)(38,59)(41,50,44,62);(1,3,7,16)(2,8,12,30)(4,13)(5,23,24,58)(6,15,26,49)(9,31)(10,35,38,45)(11,29,39,48)(14,25,47,59)(17,50)(18,41)(19,54)(20,43)(21,44)(22,57,37,28)(27,34)(32,61)(33,60)(36,63)(40,51)(42,53)(46,55)(52,62)(56,64);(1,4)(2,9)(3,13)(5,19)(6,18)(7,21)(8,27)(10,36)(11,32)(12,34)(14,46)(15,41)(16,44)(17,38)(20,26)(22,51)(23,54)(24,56)(25,53)(28,40)(29,60)(30,31)(33,39)(35,50)(37,52)(42,47)(43,49)(45,63)(48,61)(55,59)(57,62)(58,64)

Polycyclic presentation:

2^9|a^2=f,b^2=dfi,e^2=hi,g^2=i,b^a=bd,c^a=ce,e^a=eghi,g^a=gh,h^a=hi,c^b=ch,e^b=egh,h^b=hi,d^c=dg,e^c=ehi,f^c=fgi,g^c=gi,e^d=eh,g^d=gi,f^e=fh,g^f=gi

Conjugacy classes & character table

35 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72279336, #72279632, #72279992, …
maximal normal subgroup of	#72279336, #72279632, #72279992, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59807);