#512678

order	512
nilpotency class	5
isoclinism	isoclinic class of #512666
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#2127
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₄⋊Q₈	C₂⁴
prime radical	#60820	C₂
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂×#41 ◁ #3499 ◁ G
LECS	G ▷ C₄⋊Q₈ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

815 conjugacy classes of subgroups; only the 184 normal subgroups are shown.

normal subgroups only

This group has 815 conjugacy classes of subgroups — too many to draw. Only its 184 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419874);