#139385

order	512
nilpotency class	4
isoclinism	isoclinic class of #139348
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	C₂³	C₂×#102
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

317 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 317 conjugacy classes of subgroups — too many to draw. Only its 86 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 46581);