#138668

order	512
nilpotency class	4
isoclinism	isoclinic class of #12465
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1677
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂×#135	D₄
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

357 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 357 conjugacy classes of subgroups — too many to draw. Only its 96 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 — 3 generators on 68 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfghi,d^2=gh,e^2=hi,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,d^b=dh,e^b=ehi,f^b=fh,h^b=hi,d^c=dhi,e^c=eh,f^c=fhi,h^c=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, 45864);