#137263

order	512
nilpotency class	4
isoclinism	isoclinic class of #137149
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1874
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂⁴	#102
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#431 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

373 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 373 conjugacy classes of subgroups — too many to draw. Only its 82 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)(3,10,14,16)(4,13,15,7)(5,8)(6,9)(11,12)(17,18,21,26,38,40,36,25,23,28,35,45,22,27,20,34)(19,29,31,24,39,41,30,46,33,47,48,43,32,44,37,42);(1,3,11,13,6,14,5,7)(2,4,12,10,9,15,8,16)(17,19,22,39,23,33,38,32)(18,24,40,42,28,43,27,46)(20,31,35,30,36,48,21,37)(25,44,45,47,34,41,26,29);(1,4)(2,7)(3,12)(5,16)(6,15)(8,14)(9,13)(10,11)

Polycyclic presentation:

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

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, 44459);