#464499

order	512
nilpotency class	3
isoclinism	isoclinic class of #455107
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

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

934 conjugacy classes of subgroups; only the 218 normal subgroups are shown.

normal subgroups only

This group has 934 conjugacy classes of subgroups — too many to draw. Only its 218 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)(5,6,10,8)(7,12,9,11)(13,14,17,21)(15,24,27,19)(16,29,31,39)(18,23,25,22)(20,33,38,32)(26,37,42,44)(28,35,41,36)(30,40,43,34);(5,7,10,9)(6,11,8,12)(13,15,25,41,17,27,18,28)(14,19,22,35,21,24,23,36)(16,30,33,26,31,43,32,42)(20,37,29,34,38,44,39,40);(5,8,10,6)(7,11,9,12)(13,16,17,31)(14,20,21,38)(15,26,27,42)(18,33,25,32)(19,34,24,40)(22,39,23,29)(28,43,41,30)(35,37,36,44);(2,4)(5,9,10,7)(6,12,8,11)(16,31)(20,38)(26,42)(29,39)(30,43)(32,33)(34,40)(37,44)

Polycyclic presentation:

2^9|a^2=e,b^2=gi,c^2=e,d^2=ei,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, 371695);