#142886

order	512
nilpotency class	4
isoclinism	isoclinic class of #142798
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2021
commutator, Frattini, cosocle	C₂×#117	C₂³
prime radical	#3449	C₂²
LCS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

582 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 582 conjugacy classes of subgroups — too many to draw. Only its 72 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:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=gi,d^2=g,e^2=i,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=ei,f^a=fhi,c^b=cf,d^b=dg,f^b=fg,h^b=hi,d^c=dh,e^c=ei,f^c=fg,f^d=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, 50082);