#108568

order	512
nilpotency class	3
isoclinism	isoclinic class of #108562
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

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	#117
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

431 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 431 conjugacy classes of subgroups — too many to draw. Only its 142 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=d,d^2=hi,f^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,g^a=ghi,f^b=fi,e^c=eh

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, 15764);