#144126

order	512
nilpotency class	4
isoclinism	isoclinic class of #144123
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#493
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂×#493	C₂²
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

547 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 547 conjugacy classes of subgroups — too many to draw. Only its 120 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 51322);