#137248

order	512
nilpotency class	4
isoclinism	isoclinic class of #137153
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	C₂³×C₈	C₂³
prime radical	C₂⁴	#137
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#424 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 365 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)(3,6)(4,11,12,7)(5,8)(9,13,15,16)(10,14)(17,18,23,29,22,28,39,27,25,30,38,41,24,19,21,32)(20,31,35,46,40,26,37,44,36,47,42,43,34,45,33,48);(1,3,5,10)(2,6,8,14)(4,9,12,15)(7,13,11,16)(17,19,22,30,25,28,24,18)(20,31,40,45,36,47,34,26)(21,32,23,41,39,27,38,29)(33,46,35,48,37,43,42,44);(1,4,5,12)(2,7,8,11)(3,9,10,15)(6,13,14,16)(17,20,25,36)(18,26,30,45)(19,31,28,47)(21,37,39,33)(22,40,24,34)(23,42,38,35)(27,46,32,43)(29,48,41,44)

Polycyclic presentation:

2^9|a^2=f,b^2=dfgh,c^2=h,d^2=g,e^2=i,f^2=gi,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ehi,d^b=dg,e^b=eh,f^b=fg,g^b=gi,d^c=dh,e^c=ei,f^c=fh

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 44444);