#489113

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	#61284	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 708 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=hi,b^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, 396309);