#139489

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	C₂²×#41	D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#536 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 498 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=i,d^2=gh,e^2=i,f^2=gh,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, 46685);