#105840

order	512
nilpotency class	3
isoclinism	isoclinic class of #105838
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	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

1736 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 1736 conjugacy classes of subgroups — too many to draw. Only its 202 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)(3,7,8,5)(4,6)(9,10)(11,19)(12,23,25,31)(13,17)(14,18)(15,21)(16,24,30,26)(20,35,36,40)(22,34)(27,32)(28,39,33,37)(29,38);(1,3)(2,5)(4,8)(6,7)(9,11)(10,15,18,19)(12,20)(13,22)(14,21)(16,28,30,33)(17,29,32,34)(23,40,31,35)(24,37)(25,36)(26,39)(27,38);(9,12,14,25)(10,16,18,30)(11,20,21,36)(13,26,27,24)(15,28,19,33)(17,31,32,23)(22,39,38,37)(29,35,34,40)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=i,e^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=gi,d^b=di,f^b=fh,e^c=ei,f^c=fi

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, 13036);