#479581

order	512
nilpotency class	3
isoclinism	isoclinic class of #479580
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#17216
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 839 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)(3,13)(4,18)(5,22)(6,25)(7,11)(8,16)(9,20)(10,23)(12,27)(14,31)(15,41)(17,30)(19,49)(21,33)(24,35)(26,34)(28,39)(29,46)(32,53)(36,44)(37,45)(38,60)(40,57)(42,64)(43,54)(47,51)(48,52)(50,55)(56,63)(58,61)(59,62);(1,3)(2,8,11,13)(4,14)(5,15)(6,17)(7,16)(9,31,18,28)(10,29,34,36)(12,30,25,37)(19,42)(20,39)(21,40)(22,54,35,41)(23,44)(24,43)(26,46)(27,45)(32,56,47,58)(33,59,48,57)(38,50)(49,60,55,64)(51,63)(52,62)(53,61);(1,4)(2,9,11,18)(3,14)(5,19)(6,21)(7,20)(8,28,13,31)(10,32,34,47)(12,33,25,48)(15,42)(16,39)(17,40)(22,55,35,49)(23,51)(24,50)(26,53)(27,52)(29,58,36,56)(30,57,37,59)(38,43)(41,64,54,60)(44,63)(45,62)(46,61);(1,5,7,24)(2,10,11,34)(3,15,16,43)(4,19,20,50)(6,26,27,23)(8,29,13,36)(9,32,18,47)(12,35,25,22)(14,38,39,42)(17,46,45,44)(21,53,52,51)(28,56,31,58)(30,41,37,54)(33,49,48,55)(40,63,62,61)(57,60,59,64)

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285237, #72286082, #72286106
maximal normal subgroup of	#72285237, #72286082, #72286106

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 386777);