#144035

order	512
nilpotency class	4
isoclinism	isoclinic class of #143957
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	C₂×C₄×C₈	C₂³
prime radical	C₂³	#469
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

330 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 84 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) = 256

A minimal faithful permutation representation — 3 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=g,d^2=g,e^2=hi,f^2=gi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fg,h^a=hi,c^b=cf,d^b=dg,f^b=fgi,d^c=dg,e^c=eh,f^c=fgi,h^c=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 51231);