#490899

order	512
nilpotency class	4
isoclinism	isoclinic class of #490891
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#459
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#33780	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

457 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 457 conjugacy classes of subgroups — too many to draw. Only its 128 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 398095);