#472025

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	C₂×#532	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1352 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 1352 conjugacy classes of subgroups — too many to draw. Only its 202 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:

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

Polycyclic presentation:

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

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, 379221);