#152178

order	512
nilpotency class	5
isoclinism	isoclinic class of #152176
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#12731
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	#62635	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#131 ◁ C₂×#504 ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

615 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 615 conjugacy classes of subgroups — too many to draw. Only its 104 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:

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

Polycyclic presentation:

2^9|a^2=f,d^2=ghi,e^2=i,f^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gh,h^a=hi,d^b=dgi,f^b=fi,g^b=gh,h^b=hi,e^c=ei,f^c=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225560, #72225566, #72225572, …
maximal normal subgroup of	#72225560, #72225566, #72225572, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59374);