#94777

order	512
nilpotency class	6
isoclinism	isoclinic class of #7519
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1402
commutator	C₄⋊Q₈	C₄²
Frattini, cosocle	C₂×#494	C₂²
prime radical	C₂³	AΣL(1,𝔽₁₆)
LCS	G ▷ C₄⋊Q₈ ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×#494 ◁ G
LECS	G ▷ C₂×#494 ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

207 conjugacy classes of subgroups; only the 38 normal subgroups are shown.

normal subgroups only

This group has 207 conjugacy classes of subgroups — too many to draw. Only its 38 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) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 2 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=i,d^2=i,f^2=hi,g^2=i,b^a=bc,c^a=cf,e^a=ei,f^a=fg,g^a=gh,h^a=hi,d^b=dfh,f^b=fhi,g^b=ghi,d^c=dgh,f^c=fhi,g^c=gi,f^d=fhi,g^d=gi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1973);