#142848

order	512
nilpotency class	4
isoclinism	isoclinic class of #142801
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	#18300	C₂
LCS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

618 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 618 conjugacy classes of subgroups — too many to draw. Only its 68 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=gi,d^2=g,e^2=i,f^2=gi,b^a=bd,c^a=ce,d^a=dg,e^a=ei,f^a=fhi,c^b=cf,d^b=dg,f^b=fgi,h^b=hi,d^c=dh,e^c=ei,f^c=fgi,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, 50044);