#479835

order	512
nilpotency class	3
isoclinism	isoclinic class of #479834
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#17272
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

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

normal subgroups only

This group has 755 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)(3,8,15,32)(4,17)(5,23)(6,27)(7,11)(9,22)(10,25)(12,19)(13,39,18,49)(14,30,44,59)(16,46,45,31)(20,48)(21,38)(24,35,55,53)(26,37)(28,34)(29,47,58,41)(33,51)(36,56)(40,61,43,60)(42,54)(50,64,63,57)(52,62);(1,3)(2,8,11,32)(4,18)(5,24)(6,16)(7,15)(9,29,34,58)(10,35,23,53)(12,31,27,46)(13,21)(14,42)(17,39,38,49)(19,45)(20,50)(22,47)(25,55)(26,44)(28,41)(30,54,59,37)(33,60,36,61)(40,56)(43,51)(48,64,62,57)(52,63);(1,4,19,22,7,21,6,28)(2,9,12,38,11,34,27,17)(3,13,16,47,15,18,45,41)(5,20,26,56,25,52,42,51)(8,29,46,39,32,58,31,49)(10,36,54,62,23,33,37,48)(14,43,55,63,44,40,24,50)(30,57,53,61,59,64,35,60);(1,5,6,26,7,25,19,42)(2,10,12,37,11,23,27,54)(3,14,16,24,15,44,45,55)(4,20,22,51,21,52,28,56)(8,30,31,53,32,59,46,35)(9,33,17,48,34,36,38,62)(13,40,41,50,18,43,47,63)(29,57,39,60,58,64,49,61)

Polycyclic presentation:

2^9|a^2=eg,b^2=eh,c^2=efi,d^2=f,f^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,h^b=hi,d^c=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,g^d=gi,h^d=hi

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, 387031);