#105629

order	512
nilpotency class	3
isoclinism	isoclinic class of #105616
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²⋊#95
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₂×C₄⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

739 conjugacy classes of subgroups; only the 184 normal subgroups are shown.

normal subgroups only

This group has 739 conjugacy classes of subgroups — too many to draw. Only its 184 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=h,d^2=hi,f^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gi,d^b=dh,f^b=fi,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, 12825);