#121243

order	512
nilpotency class	3
isoclinism	isoclinic class of #121242
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1443
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₂²⋊Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

342 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 106 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 28439);