#503287

order	512
nilpotency class	4
isoclinism	isoclinic class of #501046
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

764 conjugacy classes of subgroups; only the 198 normal subgroups are shown.

normal subgroups only

This group has 764 conjugacy classes of subgroups — too many to draw. Only its 198 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=ei,c^2=fh,d^2=e,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=cg,d^c=di

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 410483);