#94385

order	512
nilpotency class	5
isoclinism	isoclinic class of #7408
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	AΣL(1,𝔽₁₆)
commutator	C₂×C₄⋊C₄	C₄²
Frattini, cosocle	C₄×C₄⋊Q₈	C₂²
socle	C₂²	#1390
prime radical	C₂³	#342
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₄×C₄⋊Q₈ ◁ G
LECS	G ▷ C₄×C₄⋊Q₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

222 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 46 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) = 64

A minimal faithful permutation representation — 2 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=f,d^2=fi,e^2=f,g^2=i,b^a=bc,c^a=cf,e^a=efgi,g^a=gi,h^a=hi,c^b=cg,g^b=gh,h^b=hi,e^c=ehi,g^c=gi,g^e=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1581);