#106772

order	512
nilpotency class	3
isoclinism	isoclinic class of #106766
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²⋊#95
commutator	C₂⁴	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

1766 conjugacy classes of subgroups; only the 204 normal subgroups are shown.

normal subgroups only

This group has 1766 conjugacy classes of subgroups — too many to draw. Only its 204 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)(3,7)(4,6)(5,8)(9,10,13,18,14,17,28,32)(11,19,22,33,21,15,26,30)(12,23,25,34,24,16,27,31)(20,35,37,40,36,29,38,39);(1,3)(2,5,6,7)(4,8)(9,11)(10,15)(12,20)(13,26)(14,21)(16,29)(17,19)(18,33)(22,28)(23,35)(24,36)(25,38)(27,37)(30,32)(31,40)(34,39);(9,12)(10,16)(11,20)(13,27)(14,24)(15,29)(17,23)(18,34)(19,35)(21,36)(22,38)(25,28)(26,37)(30,40)(31,32)(33,39)

Polycyclic presentation:

2^9|a^2=f,b^2=g,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gi,d^b=di,e^b=eh,f^b=fh,d^c=dh,f^c=fh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 13968);