#138327

order	512
nilpotency class	4
isoclinism	isoclinic class of #138028
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	C₂²×#41	D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#520 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 513 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:

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

Polycyclic presentation:

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

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, 45523);