#108631

order	512
nilpotency class	3
isoclinism	isoclinic class of #8746
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#381
commutator	C₂⁴	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
socle	C₂²	#1434
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

698 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 698 conjugacy classes of subgroups — too many to draw. Only its 150 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)(11,19,21,15)(12,23)(13,18)(14,17)(16,24)(20,27,33,31)(22,32,35,28)(25,36)(26,30)(29,37)(34,38,40,39);(9,11,13,22,14,21,26,35)(10,15,18,28,17,19,30,32)(12,20,25,34,24,33,37,40)(16,27,29,38,23,31,36,39);(1,3,4,8)(2,5,6,7)(9,12)(10,16,17,23)(11,20)(13,25)(14,24)(15,27,19,31)(18,29,30,36)(21,33)(22,34)(26,37)(28,38,32,39)(35,40)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=di,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^b=fh,e^c=eh,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, 15827);