#142142

order	512
nilpotency class	4
isoclinism	isoclinic class of #142050
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1993
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#418
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1726 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

298 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 298 conjugacy classes of subgroups — too many to draw. Only its 98 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=defh,c^2=eg,d^2=gh,e^2=i,f^2=h,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=fhi,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, 49338);