#134974

order	512
nilpotency class	4
isoclinism	isoclinic class of #12463
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1675
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#8644
prime radical	C₂×#135	D₄
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

285 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 285 conjugacy classes of subgroups — too many to draw. Only its 104 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 — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dg,d^2=gi,e^2=h,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,h^a=hi,e^c=ehi,f^c=fi,h^c=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42170);