#489154

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	C₂³	#447
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

658 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 658 conjugacy classes of subgroups — too many to draw. Only its 156 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)(3,9,11,16)(4,12,14,7)(6,13,15,10)(18,25)(19,30)(20,34)(21,37)(22,23)(27,41)(28,38)(29,35)(31,36)(32,39)(33,48)(40,44)(43,47)(45,46);(1,3,5,11)(2,6,8,15)(4,13,14,10)(7,16,12,9)(17,18)(19,26)(20,35)(21,38)(22,29)(23,28)(24,27)(25,34)(30,47)(31,46)(32,45)(33,42)(36,44)(37,41)(39,43)(40,48);(17,19,24,33)(18,26,27,42)(20,36,21,39)(22,32,23,31)(25,40,41,47)(28,46,29,45)(30,34,48,37)(35,44,38,43);(18,27)(25,41)(26,42)(28,29)(35,38)(40,47)(43,44)(45,46)

Polycyclic presentation:

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

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, 396350);