#140197

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	C₂²×D₈	C₂×C₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 263 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,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, 47393);