#105923

order	512
nilpotency class	3
isoclinism	isoclinic class of #105908
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	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₂²⋊Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

803 conjugacy classes of subgroups; only the 204 normal subgroups are shown.

normal subgroups only

This group has 803 conjugacy classes of subgroups — too many to draw. Only its 204 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 13119);