#144684

order	512
nilpotency class	4
isoclinism	isoclinic class of #142236
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#447
commutator, Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#408 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

444 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 74 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) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=i,c^2=d,e^2=gi,f^2=i,g^2=i,b^a=bd,c^a=ce,e^a=eg,f^a=fh,g^a=gi,c^b=cf,e^b=eh,f^b=fi,e^c=eg,f^c=fi,g^c=gi

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, 51880);