#479302

order	512
nilpotency class	3
isoclinism	isoclinic class of #479301
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#17198
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 727 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)(3,8,15,28)(4,9)(5,20,21,51)(6,12)(7,11)(10,22,36,52)(13,30,41,26)(14,43)(16,29,23,38)(17,48,49,60)(18,33)(19,32)(24,39)(25,37)(27,53)(31,50,35,54)(34,46)(40,61)(42,57,47,56)(44,63)(45,59)(55,62)(58,64);(1,3)(2,8,11,28)(4,13)(5,14)(6,23)(7,15)(9,30,32,26)(10,34,36,27)(12,38,37,29)(16,25)(17,40)(18,47)(19,41)(20,43,51,59)(21,45)(22,53)(24,42)(31,55,35,58)(33,56,39,57)(44,49)(46,52)(48,61,60,63)(50,64)(54,62);(1,4,7,19)(2,9,11,32)(3,13,15,41)(5,17,21,49)(6,24,25,18)(8,26,28,30)(10,35,36,31)(12,39,37,33)(14,44,45,40)(16,47,23,42)(20,48,51,60)(22,54,52,50)(27,55,34,58)(29,57,38,56)(43,61,59,63)(46,62,53,64);(1,5)(2,10,11,36)(3,14)(4,17)(6,22)(7,21)(8,27,28,34)(9,31,32,35)(12,20,37,51)(13,40)(15,45)(16,46)(18,50)(19,49)(23,53)(24,54)(25,52)(26,55,30,58)(29,43,38,59)(33,48,39,60)(41,44)(42,62)(47,64)(56,63,57,61)

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 386498);