#140198

order	512
nilpotency class	4
isoclinism	isoclinic class of #13313
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#466
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1934
prime radical	C₂³	C₄×SD₁₆
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

247 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 247 conjugacy classes of subgroups — too many to draw. Only its 88 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfgi,c^2=d,d^2=gi,e^2=gi,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,c^b=ch,e^c=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 47394);