#134098

order	512
nilpotency class	4
isoclinism	isoclinic class of #133889
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1781
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂×#117	C₂³
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#536 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

831 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 831 conjugacy classes of subgroups — too many to draw. Only its 86 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=hi,d^2=i,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,d^b=di,f^b=fgi,e^c=ei,f^c=fh,f^e=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, 41294);