#490771

order	512
nilpotency class	4
isoclinism	isoclinic class of #490699
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	#33669	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 126 normal subgroups are shown.

normal subgroups only

This group has 457 conjugacy classes of subgroups — too many to draw. Only its 126 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)(3,10)(4,8)(5,14)(6,9)(7,13)(11,16)(12,15)(17,18,22,27)(19,23,28,36)(20,24,33,40)(21,34,32,26)(25,41,39,31)(29,35,42,46)(30,45,43,38)(37,48,47,44);(1,3,5,13,6,11,4,12)(2,7,8,10,9,15,14,16)(17,19)(18,23)(20,29)(21,30)(22,28)(24,35)(25,37)(26,38)(27,36)(31,44)(32,43)(33,42)(34,45)(39,47)(40,46)(41,48);(3,11)(7,15)(10,16)(12,13)(18,24)(20,31)(21,32)(23,35)(25,26)(27,40)(29,44)(30,43)(33,41)(34,39)(36,46)(37,38)(42,48)(45,47);(1,4,6,5)(2,8,9,14)(3,12,11,13)(7,10,15,16)(19,28)(23,36)(29,42)(30,43)(35,46)(37,47)(38,45)(44,48)

Polycyclic presentation:

2^9|a^2=i,b^2=e,d^2=g,e^2=g,f^2=hi,h^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fh,h^a=hi,c^b=cg,d^b=di,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, 397967);