#219612

order	512
nilpotency class	3
isoclinism	isoclinic class of #219600
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2328
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂³	#535
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

476 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 162 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) = 128

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egi,c^2=efi,d^2=ei,e^2=h,f^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=ce,d^b=di,f^b=fh,g^c=gh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 126808);