#134953

order	512
nilpotency class	4
isoclinism	isoclinic class of #12416
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	#459
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

328 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 328 conjugacy classes of subgroups — too many to draw. Only its 122 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 — 3 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dgi,c^2=i,d^2=g,e^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,e^c=eh,h^c=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42149);