#139951

order	512
nilpotency class	4
isoclinism	isoclinic class of #139724
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	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂×#117	C₂³
prime radical	C₂×#121	C₂×C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#542 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

419 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 419 conjugacy classes of subgroups — too many to draw. Only its 66 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=i,d^2=i,e^2=i,f^2=hi,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,g^a=gi,d^b=dg,e^b=eh,f^b=fgi,g^b=gi,d^c=dh,e^c=ei,f^c=fh,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, 47147);