#141103

order	512
nilpotency class	4
isoclinism	isoclinic class of #140973
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1961
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂×#493	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1726 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

364 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 86 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) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=egh,d^2=ghi,e^2=i,f^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=cg,d^b=dh,e^b=ei,f^b=fh,f^c=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 48299);