#139359

order	512
nilpotency class	4
isoclinism	isoclinic class of #139358
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1893
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	#525	D₄
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #2929 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

349 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 349 conjugacy classes of subgroups — too many to draw. Only its 78 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)(3,6,10,14)(4,11,12,7)(5,8)(9,13)(15,16)(18,24,28,34)(19,30)(20,33)(21,22)(25,35)(26,41,43,48)(27,39,29,38)(31,46)(32,36)(37,44,47,42);(1,3,5,10)(2,6)(4,9,12,15)(7,13)(8,14)(11,16)(17,18,25,39,21,27,35,24,23,28,33,38,22,29,20,34)(19,26,40,41,36,44,46,37,32,43,45,48,30,42,31,47);(1,4)(2,7)(3,9)(5,12)(6,13)(8,11)(10,15)(14,16)(17,19)(18,26)(20,31)(21,36)(22,30)(23,32)(24,37)(25,40)(27,44)(28,43)(29,42)(33,45)(34,47)(35,46)(38,48)(39,41)

Polycyclic presentation:

2^9|a^2=f,b^2=dfgi,d^2=ghi,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,d^b=dh,e^b=ehi,f^b=fh,d^c=dhi,e^c=ei,f^c=fhi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 46555);