#106026

order	512
nilpotency class	3
isoclinism	isoclinic class of #106017
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₄×C₂²⋊C₄
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	C₂²⋊#95
prime radical	C₂⁴	#117
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

714 conjugacy classes of subgroups; only the 190 normal subgroups are shown.

normal subgroups only

This group has 714 conjugacy classes of subgroups — too many to draw. Only its 190 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=h,e^2=i,f^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gi,d^b=di,e^b=ei,f^b=fh,d^c=di,e^c=ei,f^c=fhi

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, 13222);