#139680

order	512
nilpotency class	4
isoclinism	isoclinic class of #139568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1893
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 ◁ #2960 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

310 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 78 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 46876);