#491549

order	512
nilpotency class	4
isoclinism	isoclinic class of #491511
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	#33748	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 523 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:

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

Polycyclic presentation:

2^9|b^2=ei,c^2=fhi,d^2=gi,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

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, 398745);