#94785

order	512
nilpotency class	6
isoclinism	isoclinic class of #7523
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1404
commutator	#125	C₄²
Frattini, cosocle	C₂×#491	C₂²
prime radical	C₂³	AΣL(1,𝔽₁₆)
LCS	G ▷ #125 ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×#491 ◁ G
LECS	G ▷ C₂×#491 ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 181 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=hi,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,c^b=ch,d^b=df,f^b=fhi,g^b=gh,d^c=dgh,f^c=fh,g^c=gi,f^d=fhi,g^d=gi

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1981);