#487258

order	512
nilpotency class	4
isoclinism	isoclinic class of #486867
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×C₄⋊C₄	C₂⁴
prime radical	#392	D₄
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

688 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 688 conjugacy classes of subgroups — too many to draw. Only its 116 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=i,d^2=i,e^2=g,f^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fi,h^a=hi,d^b=di,e^b=eg,f^b=fhi,d^c=dg,e^c=eh,f^c=fi,f^e=fi

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 394454);