#491667

order	512
nilpotency class	4
isoclinism	isoclinic class of #491662
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#493
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#19667	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

527 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 527 conjugacy classes of subgroups — too many to draw. Only its 146 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)(17,18,23,27)(19,24,29,39)(20,32,35,36)(21,34,37,41)(22,33,25,26)(28,42,43,47)(30,45,46,48)(31,44,38,40);(1,2,4,6,5,8,11,13)(3,7,10,12,9,14,15,16)(17,19,23,29)(18,24,27,39)(20,28,35,43)(21,30,37,46)(22,31,25,38)(26,40,33,44)(32,42,36,47)(34,45,41,48);(1,3)(2,7)(4,10)(5,9)(6,12)(8,14)(11,15)(13,16)(17,20,32,27,22,34,21,33,23,35,36,18,25,41,37,26)(19,28,42,39,31,45,30,44,29,43,47,24,38,48,46,40);(3,9)(7,14)(10,15)(12,16)(19,29)(24,39)(28,43)(30,46)(31,38)(40,44)(42,47)(45,48)

Polycyclic presentation:

2^9|a^2=i,b^2=ei,c^2=fhi,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=dg

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 398863);