#464274

order	512
nilpotency class	3
isoclinism	isoclinic class of #455107
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1390 conjugacy classes of subgroups; only the 210 normal subgroups are shown.

normal subgroups only

This group has 1390 conjugacy classes of subgroups — too many to draw. Only its 210 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) = 256

A minimal faithful permutation representation — 4 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=g,c^2=i,d^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^c=de,f^c=fi,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 371470);