#490297

order	512
nilpotency class	4
isoclinism	isoclinic class of #490115
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	#62341	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

670 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 670 conjugacy classes of subgroups — too many to draw. Only its 134 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 397493);