#512639

order	512
nilpotency class	5
isoclinism	isoclinic class of #512636
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂²×#354
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₄⋊Q₈	C₂⁴
prime radical	#3498	C₂²
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×Q₈ ◁ C₂×#41 ◁ #3498 ◁ G
LECS	G ▷ C₄⋊Q₈ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

818 conjugacy classes of subgroups; only the 192 normal subgroups are shown.

normal subgroups only

This group has 818 conjugacy classes of subgroups — too many to draw. Only its 192 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) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=i,d^2=i,e^2=hi,f^2=i,g^2=i,b^a=be,e^a=eg,g^a=gh,h^a=hi,e^b=eh,f^b=fgh,g^b=gi,d^c=di,f^e=fhi,g^f=gi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419835);