#108680

order	512
nilpotency class	3
isoclinism	isoclinic class of #108672
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1434
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₂×C₄⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

419 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 419 conjugacy classes of subgroups — too many to draw. Only its 140 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) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2,4,8,5,6,11,14)(3,7,10,15,9,13,12,16)(17,18)(19,29)(20,34,36,41)(21,27)(22,26)(23,28)(24,31)(25,35,42,37)(30,38,45,48)(32,40)(33,39)(43,44,47,46);(2,6)(4,11)(7,13)(10,12)(17,19,22,32,23,33,21,31)(18,24,27,39,28,40,26,29)(20,30,37,46,36,45,35,44)(25,38,34,47,42,48,41,43);(1,3)(2,7,6,13)(4,12)(5,9)(8,16,14,15)(10,11)(17,20,21,35,23,36,22,37)(18,25,26,41,28,42,27,34)(19,30,31,44,33,45,32,46)(24,38,29,43,40,48,39,47)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=d,d^2=hi,e^2=hi,f^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=ghi,f^b=fi,e^c=ei,f^c=fh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 15876);