#140634

order	512
nilpotency class	4
isoclinism	isoclinic class of #13229
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1964
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂×#117	C₂³
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

383 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 383 conjugacy classes of subgroups — too many to draw. Only its 104 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=ei,d^2=i,e^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,d^b=dg,f^b=fgi,g^b=gi,f^d=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 47830);