#139037

order	512
nilpotency class	4
isoclinism	isoclinic class of #139018
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1879
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂²×D₈	C₂×C₄
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1451 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

316 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 76 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfgh,c^2=i,d^2=ghi,e^2=gh,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dh,e^b=ehi,f^b=fh,h^b=hi,d^c=dhi,e^c=eghi,f^c=fhi,h^c=hi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 46233);