#142293

order	512
nilpotency class	4
isoclinism	isoclinic class of #142242
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1999
commutator, Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#424 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

567 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 567 conjugacy classes of subgroups — too many to draw. Only its 70 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)(4,6)(7,8)(9,10,15,19)(11,20,25,39)(12,26,18,14)(13,29,32,28)(16,33,34,22)(17,35,37,31)(21,40,27,30)(23,36,24,38);(5,8)(9,11)(10,16)(12,27)(13,30)(14,23)(15,25)(17,36)(18,21)(19,34)(20,29)(22,31)(24,26)(28,39)(32,40)(33,35)(37,38);(1,3)(2,5)(4,7)(6,8)(9,12,14,10,15,18,26,19)(11,21,24,16,25,27,23,34)(13,29,31,37,32,28,35,17)(20,22,36,30,39,33,38,40)

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 49489);