#219637

order	512
nilpotency class	3
isoclinism	isoclinic class of #219624
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)(3,14)(4,18,22,27)(5,17)(6,12)(7,13)(8,10)(9,21,28,32)(11,23)(15,29,31,24)(16,25)(19,26,30,20)(33,34)(35,38)(36,47,50,40)(37,51)(39,43)(41,52)(42,44)(45,49,55,57)(46,53)(48,59,62,63)(54,61,60,64)(56,58);(1,3,6,10,7,17,23,16)(2,8,11,14,13,25,12,5)(4,19,32,15,22,30,21,31)(9,26,18,24,28,20,27,29)(33,35,44,43,39,42,38,34)(36,40,49,55,50,47,57,45)(37,46,58,56,53,51,52,41)(48,54,61,63,62,60,64,59);(1,4,7,22)(2,9)(3,15)(5,20,14,26)(6,21,23,32)(8,24,25,29)(10,19)(11,27)(12,18)(13,28)(16,30)(17,31)(33,36,44,57,39,50,38,49)(34,40,42,55,43,47,35,45)(37,48,58,64,53,62,52,61)(41,54,51,63,56,60,46,59);(1,5,23,25,7,14,6,8)(2,10,12,3,13,16,11,17)(4,20,32,29,22,26,21,24)(9,19,18,15,28,30,27,31)(33,37,38,52,39,53,44,58)(34,41,42,51,43,56,35,46)(36,48,49,61,50,62,57,64)(40,54,55,63,47,60,45,59)

Polycyclic presentation:

2^9|a^2=g,b^2=egi,c^2=efi,d^2=ehi,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=dhi,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, 126833);