#483101

order	512
nilpotency class	4
isoclinism	isoclinic class of #483101
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂²×#333
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	C₂×#41	C₂²⋊C₄
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×#41 ◁ C₄×#131 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

705 conjugacy classes of subgroups; only the 312 normal subgroups are shown.

normal subgroups only

This group has 705 conjugacy classes of subgroups — too many to draw. Only its 312 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 — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,e^2=hi,f^2=gh,g^2=i,b^a=be,e^a=eg,g^a=gi,h^a=hi,e^b=eh,f^b=fgh,g^b=gi,d^c=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390297);