#477176

order	512
nilpotency class	3
isoclinism	isoclinic class of #477175
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#17105
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1081 conjugacy classes of subgroups; only the 168 normal subgroups are shown.

normal subgroups only

This group has 1081 conjugacy classes of subgroups — too many to draw. Only its 168 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:

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

Polycyclic presentation:

2^9|a^2=efi,b^2=f,c^2=e,d^2=e,f^2=i,g^2=i,h^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,g^b=gi,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, 384372);