#131364

order	512
nilpotency class	4
isoclinism	isoclinic class of #12108
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

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	C₂×#466	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

657 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 657 conjugacy classes of subgroups — too many to draw. Only its 106 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)(7,8,12,10)(9,11)(13,14,18,21,20,24,16,31,19,25,32,38,17,35,34,23)(15,26,29,42,28,22,33,37,30,41,44,39,36,40,27,43);(2,4)(5,7)(6,10)(8,11)(9,12)(14,21)(16,32)(17,20)(18,34)(22,37)(23,24)(25,38)(26,39)(27,29)(28,36)(31,35)(33,44)(40,43)(41,42);(5,8,9,10)(6,7,11,12)(13,15)(14,22)(16,33)(17,36)(18,29)(19,30)(20,28)(21,37)(23,42)(24,41)(25,40)(26,35)(27,34)(31,39)(32,44)(38,43)

Polycyclic presentation:

2^9|a^2=f,c^2=gi,d^2=hi,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dh,e^b=eg,f^b=fgh,h^b=hi,d^c=dg,e^c=ei,f^c=fgi

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, 38560);