#94339

order	512
nilpotency class	5
isoclinism	isoclinic class of #7418
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#357
commutator	C₂×C₁₆	C₄²
Frattini, cosocle	C₂³×C₁₆	C₂²
prime radical	C₂⁴	D₁₆
LCS	G ▷ C₂×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×C₁₆ ◁ G
LECS	G ▷ C₂³×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

264 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 86 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) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 2 generators on 44 points:

(1,2,3,4)(5,6)(7,9)(8,11)(10,12)(13,14)(15,20)(16,31)(17,33)(18,35)(19,22)(21,28)(23,27)(24,34)(25,32)(26,40)(29,41)(30,36)(37,44)(38,43)(39,42);(6,9,11,10)(14,20,22,36)(15,26)(16,27)(17,34)(18,32)(21,24,40,23)(25,41,35,39)(28,30)(29,44)(31,37,33,38)(42,43)

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fh,f^2=hi,g^2=hi,h^2=i,b^a=bc,c^a=cf,e^a=efgh,f^a=fh,g^a=gh,h^a=hi,c^b=cg,f^b=fh,g^b=gh,h^b=hi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1535);