#94352

order	512
nilpotency class	5
isoclinism	isoclinic class of #7418
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

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

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

normal subgroups only

This group has 160 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) = 512

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

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

Polycyclic presentation:

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

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, 1548);