#479731

order	512
nilpotency class	3
isoclinism	isoclinic class of #478580
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2789
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#17230
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 549 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)(3,13,16,8)(4,18)(5,23,25,53)(6,12)(7,11)(9,21)(10,26,35,55)(14,29,19,38)(15,43)(17,40,45,31)(20,48,49,52)(22,37)(24,51)(27,33)(28,36)(30,44)(32,56,34,50)(39,54)(41,61)(42,58,46,60)(47,63)(57,64)(59,62);(1,3)(2,8)(4,19,21,14)(5,24)(6,17)(7,16)(9,29,18,38)(10,30)(11,13)(12,31)(15,25)(20,41,49,47)(22,46,27,42)(23,51)(26,54)(28,45)(32,57,34,59)(33,58,37,60)(35,39)(36,40)(43,53)(44,55)(48,63,52,61)(50,62,56,64);(1,4)(2,9,11,18)(3,14)(5,20,25,49)(6,27)(7,21)(8,29,13,38)(10,34)(12,37,36,33)(15,41,24,47)(16,19)(17,46)(22,28)(23,52)(26,56,55,50)(30,59)(31,60,40,58)(32,35)(39,57)(42,45)(43,63)(44,64,54,62)(48,53)(51,61);(1,5)(2,10,11,35)(3,15,16,24)(4,20,21,49)(6,26)(7,25)(8,30)(9,32)(12,23,36,53)(13,39)(14,41)(17,44,45,54)(18,34)(19,47)(22,50,27,56)(28,55)(29,57,38,59)(31,51)(33,52)(37,48)(40,43)(42,62)(46,64)(58,61,60,63)

Polycyclic presentation:

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

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, 386927);