#140973

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1961
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂×#466	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

440 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 82 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)(3,5,7,8)(4,6)(9,10)(11,22)(12,27,31,36)(13,25)(14,19)(15,21)(16,20)(17,30,35,33)(18,23)(24,29,34,39)(26,32)(28,37,40,38);(1,3,4,7)(2,5)(6,8)(9,11,23,21,14,26,32,19,15,18,22,10,16,25,13,20)(12,28,37,36,30,34,29,35,31,40,38,27,33,24,39,17);(2,6)(3,7)(9,12,15,31)(10,17)(11,24)(13,29,32,39)(14,30,16,33)(18,34)(19,27)(20,36)(21,35)(22,37,23,38)(25,40)(26,28)

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=eg,d^2=ghi,e^2=i,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, 48169);