#134772

order	512
nilpotency class	4
isoclinism	isoclinic class of #134769
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1808
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	#525	D₄
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #2929 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 313 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)(3,7,8,5)(4,6)(10,16)(11,22)(12,26)(13,14)(17,27)(18,34)(19,31)(20,30)(21,32)(23,38)(24,40)(25,28)(29,36)(37,39);(9,10,17,30,14,20,12,16,15,21,26,31,13,19,27,32)(11,23,33,34,22,37,24,39,25,18,35,38,28,36,40,29);(1,3)(2,5)(4,8)(6,7)(9,11)(10,18)(12,24)(13,28)(14,22)(15,25)(16,29)(17,33)(19,37)(20,36)(21,23)(26,35)(27,40)(30,38)(31,34)(32,39)

Polycyclic presentation:

2^9|a^2=f,b^2=dgi,d^2=gi,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,c^b=ci,e^c=ei,f^c=fhi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 41968);