#141682

order	512
nilpotency class	4
isoclinism	isoclinic class of #12506
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	#493
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

416 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 142 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)(15,29)(16,22)(17,36)(18,39)(19,27)(20,26)(21,28)(23,33)(24,37)(25,38)(30,41)(31,40)(32,44)(34,42)(35,43);(2,4)(5,6,10,12,8,11,7,9)(13,15,18,30,19,31,17,16,21,33,37,34,20,32,38,35)(14,22,25,40,26,41,24,29,28,43,36,44,27,42,39,23);(13,16,17,32,20,30,18,33,21,35,38,31,19,34,37,15)(14,23,24,41,27,44,25,22,28,29,39,42,26,40,36,43)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 48878);