#93556

order	512
nilpotency class	3
isoclinism	isoclinic class of #93508
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	H(3,ℤ/4)
commutator	C₂²×C₈	C₄²
Frattini, cosocle	C₂×C₄×M₁₆	C₂²
socle	C₂²	#1288
prime radical	C₂⁴	#104
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂×C₄×M₁₆ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

294 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 64 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 — 2 generators on 40 points:

(2,6)(3,4)(7,8)(9,10)(11,20,22,36)(12,25,28,39)(13,29,30,17)(14,18)(15,19)(16,21)(23,33)(24,31,37,40)(26,32)(27,34)(35,38);(1,2,4,7,5,8,3,6)(9,11,13,23,27,37,12,26,15,22,30,21,14,24,28,38)(10,16,17,31,34,35,39,20,19,33,29,40,18,32,25,36)

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fhi,e^2=fi,f^2=i,g^2=i,b^a=bc,c^a=cf,e^a=efghi,h^a=hi,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei,e^d=ei

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, 752);