#479650

order	512
nilpotency class	3
isoclinism	isoclinic class of #476143
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2819
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#17222
prime radical	C₂⁴	C₂²×Q₈
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

555 conjugacy classes of subgroups; only the 160 normal subgroups are shown.

normal subgroups only

This group has 555 conjugacy classes of subgroups — too many to draw. Only its 160 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)(3,8,15,27)(4,9,19,31)(5,10,21,34)(6,12)(7,11)(13,25)(14,26)(16,28,36,49)(17,29)(18,30)(20,32,43,50)(22,35,39,51)(23,24)(33,48)(37,52,40,63)(38,53)(41,55)(42,54)(44,56,46,58)(45,57)(47,59)(60,62)(61,64);(1,3,6,16,7,15,23,36)(2,8,24,49,11,27,12,28)(4,17,43,13,19,45,20,38)(5,14,39,61,21,41,22,42)(9,29,32,53,31,57,50,25)(10,26,35,54,34,55,51,64)(18,46,47,37,48,44,62,40)(30,58,60,63,33,56,59,52);(1,4,6,20,7,19,23,43)(2,9,24,50,11,31,12,32)(3,13,36,45,15,38,16,17)(5,18,22,47,21,48,39,62)(8,25,28,29,27,53,49,57)(10,33,51,59,34,30,35,60)(14,40,61,44,41,37,42,46)(26,52,54,56,55,63,64,58);(1,5)(2,10,11,34)(3,14,15,41)(4,18)(6,22)(7,21)(8,26)(9,30,31,33)(12,35,24,51)(13,37,38,40)(16,42,36,61)(17,44,45,46)(19,48)(20,47)(23,39)(25,52)(27,55)(28,54)(29,56)(32,59,50,60)(43,62)(49,64)(53,63)(57,58)

Polycyclic presentation:

2^9|a^2=ef,b^2=fg,c^2=eg,d^2=e,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=def,e^b=ei,d^c=de,f^c=fi,f^d=fi,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, 386846);