#512307

order	512
nilpotency class	4
isoclinism	isoclinic class of #511893
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

589 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 589 conjugacy classes of subgroups — too many to draw. Only its 156 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) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 68 points:

(2,3)(5,6,11,16)(7,17,22,48)(8,23,25,57)(9,27,18,14)(10,30,32,58)(12,34,37,20)(13,38,40,26)(15,42,44,29)(19,49,51,68)(21,43,31,36)(24,59,28,45)(33,41,61,39)(35,55,66,64)(46,60,65,52)(47,62,63,54)(50,56,53,67);(1,2,4,3)(5,7,18,34,11,22,9,20)(6,12,27,48,16,37,14,17)(8,19,24,50,25,51,28,53)(10,31,42,54,32,21,29,62)(13,35,39,65,40,66,41,46)(15,43,58,47,44,36,30,63)(23,56,45,49,57,67,59,68)(26,60,33,64,38,52,61,55);(10,32)(15,44)(21,31)(23,57)(26,38)(29,42)(30,58)(33,61)(36,43)(45,59)(47,63)(49,68)(52,60)(54,62)(55,64)(56,67);(1,3,4,2)(5,8,11,25)(6,13,16,40)(7,19,22,51)(9,28,18,24)(10,33,32,61)(12,35,37,66)(14,41,27,39)(15,45,44,59)(17,46,48,65)(20,53,34,50)(21,55,31,64)(23,58,57,30)(26,42,38,29)(36,68,43,49)(47,67,63,56)(52,62,60,54)

Polycyclic presentation:

2^9|a^2=i,b^2=efi,d^2=eh,e^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,g^a=gh,h^a=hi,g^b=gi,g^c=gi,e^d=ei,f^d=fi,g^d=gh,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419503);