#137187

order	512
nilpotency class	4
isoclinism	isoclinic class of #12108
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#447
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂³×D₈	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#424 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

601 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 601 conjugacy classes of subgroups — too many to draw. Only its 94 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=h,d^2=gi,e^2=i,f^2=gi,g^2=i,b^a=bd,c^a=ce,e^a=ehi,d^b=dg,e^b=eh,f^b=fg,g^b=gi,d^c=dh,e^c=ei,f^c=fh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 44383);