#404687

order	512
nilpotency class	3
isoclinism	isoclinic class of #404687
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14899
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

756 conjugacy classes of subgroups; only the 168 normal subgroups are shown.

normal subgroups only

This group has 756 conjugacy classes of subgroups — too many to draw. Only its 168 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=ef,c^2=e,d^2=g,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,f^b=fi,f^c=fi,h^d=hi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 311883);