#479671

order	512
nilpotency class	3
isoclinism	isoclinic class of #477321
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2830
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#17224
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

819 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 819 conjugacy classes of subgroups — too many to draw. Only its 152 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)(3,8,15,30)(4,9,18,34)(5,20,23,40)(6,12)(7,11)(10,24,37,26)(13,28)(14,44)(16,31,46,58)(17,49)(19,35,25,39)(21,53)(22,54)(27,38)(29,48)(32,42)(33,52)(36,45)(41,61,50,64)(43,56)(47,59)(51,60)(55,62,57,63);(1,3)(2,8,11,30)(4,13)(5,21,23,14)(6,16)(7,15)(9,32,34,28)(10,36)(12,31,38,58)(17,50,22,41)(18,42)(19,43)(20,53)(24,48,26,45)(25,47)(27,46)(29,37)(33,55)(35,59,39,56)(40,44)(49,61)(51,63,52,62)(54,64)(57,60);(1,4)(2,9,11,34)(3,13)(5,22,23,17)(6,25)(7,18)(8,28,30,32)(10,33)(12,39,38,35)(14,41,21,50)(15,42)(16,47)(19,27)(20,54)(24,51,26,52)(29,57)(31,59,58,56)(36,55)(37,60)(40,49)(43,46)(44,61)(45,63,48,62)(53,64);(1,5)(2,10)(3,14)(4,17)(6,26)(7,23)(8,29)(9,33)(11,37)(12,40)(13,41)(15,21)(16,48)(18,22)(19,52)(20,38)(24,27)(25,51)(28,55)(30,36)(31,53)(32,57)(34,60)(35,54)(39,49)(42,50)(43,63)(44,58)(45,46)(47,62)(56,64)(59,61)

Polycyclic presentation:

2^9|a^2=eg,b^2=g,c^2=g,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=def,e^b=ei,d^c=de,e^c=ei,f^c=fi,h^c=hi,g^d=gi,h^d=hi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 386867);