#489250

order	512
nilpotency class	4
isoclinism	isoclinic class of #488920
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#3147	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

702 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 702 conjugacy classes of subgroups — too many to draw. Only its 148 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) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2,5,8)(3,9,10,16)(4,12,13,7)(6,14,15,11)(18,24)(20,22)(21,33)(25,40)(26,44)(27,39)(28,31)(29,45)(32,36)(34,37)(35,47)(38,42)(41,46)(43,48);(4,13)(7,12)(9,16)(11,14)(17,18,23,27)(19,25,30,42)(20,26,22,37)(21,34,36,44)(24,32,39,33)(28,41,31,43)(29,46,35,48)(38,45,40,47);(1,3,5,10)(2,6,8,15)(4,14,13,11)(7,16,12,9)(17,19,23,30)(18,25,27,42)(20,31,22,28)(21,35,36,29)(24,38,39,40)(26,43,37,41)(32,45,33,47)(34,48,44,46);(17,20,23,22)(18,26,27,37)(19,28,30,31)(21,32,36,33)(24,34,39,44)(25,41,42,43)(29,45,35,47)(38,46,40,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=h,d^2=i,e^2=gi,f^2=hi,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eg,f^a=fhi,g^a=gi,e^b=eg,f^b=fh,g^b=gi,d^c=di,e^c=eh,f^c=fhi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 396446);