#139503

order	512
nilpotency class	4
isoclinism	isoclinic class of #139440
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1898
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×#117	C₂³
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#544 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

424 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 68 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)(17,18,22,27)(19,28)(20,25)(21,26,37,42)(23,29,40,30)(24,36)(31,38)(32,45)(33,41)(34,43)(35,48)(39,46,44,47);(1,3)(2,8)(4,11)(5,14)(6,15)(7,13)(9,16)(10,12)(17,19,20,29)(18,23,25,28)(21,35,33,47)(22,32,34,30)(24,38,42,39)(26,44,41,48)(27,40,43,45)(31,36,46,37);(1,4,6,8)(2,3,11,15)(5,12,7,16)(9,14,10,13)(17,20)(18,24)(19,29)(21,36)(22,34)(23,38)(25,42)(26,43)(27,41)(28,39)(30,32)(31,47)(33,37)(35,46)(40,48)(44,45)

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=ghi,d^2=gh,e^2=i,f^2=ghi,b^a=bd,c^a=ce,d^a=dg,e^a=eghi,g^a=gi,h^a=hi,d^b=dh,e^b=egh,f^b=fh,g^b=gi,h^b=hi,d^c=dgh,e^c=ei,f^c=fgh,f^d=fi

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 46699);