#140723

order	512
nilpotency class	4
isoclinism	isoclinic class of #2215
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×D₈
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle, prime radical	C₂³	C₄×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

356 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 126 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 47919);