#512586

order	512
nilpotency class	5
isoclinism	isoclinic class of #512579
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂²×AΣL(1,𝔽₁₆)
commutator	C₂×Q₈	C₂³×C₄
Frattini, cosocle	C₄⋊Q₈	C₂⁴
prime radical	#60391	C₂
LCS	G ▷ C₂×Q₈ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×D₄ ◁ C₂×#41 ◁ #3499 ◁ G
LECS	G ▷ C₄⋊Q₈ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

1435 conjugacy classes of subgroups; only the 192 normal subgroups are shown.

normal subgroups only

This group has 1435 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419782);