#139530

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

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	#123
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#544 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 410 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfh,c^2=gh,d^2=ghi,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=fhi,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, 46726);