#141113

order	512
nilpotency class	4
isoclinism	isoclinic class of #140881
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1964
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×#117	C₂³
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

414 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 96 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfi,c^2=ei,d^2=gh,e^2=i,f^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,h^a=hi,d^b=dh,f^b=fh,g^b=gi,h^b=hi,f^d=fi

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, 48309);