#144253

order	512
nilpotency class	4
isoclinism	isoclinic class of #144219
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1999
commutator, Frattini, cosocle	C₂³×C₈	C₂³
prime radical	#33758	C₂
LCS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

413 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 413 conjugacy classes of subgroups — too many to draw. Only its 64 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)(4,7)(5,8)(9,14)(10,13)(11,15)(12,16)(17,18)(19,29)(20,24)(21,38)(22,27)(23,28)(25,32)(26,34)(30,45)(31,43)(33,44)(35,42)(36,40)(37,41)(39,46)(47,48);(1,3)(2,6)(4,9)(5,10)(7,14)(8,13)(11,16)(12,15)(18,24)(19,30)(20,34)(21,22)(25,42)(26,27)(28,41)(29,39)(31,35)(32,33)(36,37)(38,40)(43,45)(44,48);(1,4,5,11)(2,7,8,15)(3,9,10,16)(6,12,13,14)(17,19,23,33)(18,25,28,45)(20,35,37,31)(21,30,22,32)(24,39,41,48)(26,42,40,43)(27,44,38,29)(34,47,36,46)

Polycyclic presentation:

2^9|c^2=h,d^2=gi,e^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,f^a=fhi,g^a=gi,c^b=cf,d^b=dg,e^b=eh,f^b=fi,g^b=gi,d^c=di,e^c=ei,f^c=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 51449);