#219525

order	512
nilpotency class	3
isoclinism	isoclinic class of #219525
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2328
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₈	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

550 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 550 conjugacy classes of subgroups — too many to draw. Only its 162 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 — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=ef,d^2=e,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dei,e^a=eh,c^b=cei,f^b=fh,d^c=di,g^c=gh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 126721);