#140818

order	512
nilpotency class	3
isoclinism	isoclinic class of #9758
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#390
commutator	C₄²	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1954
prime radical	C₂³	#436
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

262 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 90 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) = 256

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=eg,d^2=i,e^2=gi,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,c^b=ci,d^b=dgi,f^b=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 48014);