#138605

order	512
nilpotency class	4
isoclinism	isoclinic class of #138564
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	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#465
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

293 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 293 conjugacy classes of subgroups — too many to draw. Only its 96 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) = 256

A minimal faithful permutation representation — 3 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 45801);