#486778

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	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 659 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) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|b^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, 393974);