#486782

order	512
nilpotency class	4
isoclinism	isoclinic class of #486327
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	C₂²×C₈	C₂⁴
prime radical	#62373	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

699 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 699 conjugacy classes of subgroups — too many to draw. Only its 132 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=h,d^2=gh,e^2=gi,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fi,g^a=gi,d^b=dg,e^b=eg,f^b=fh,g^b=gi,d^c=di,e^c=eh,f^c=fi

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, 393978);