#488823

order	512
nilpotency class	4
isoclinism	isoclinic class of #488719
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 697 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:

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

Polycyclic presentation:

2^9|c^2=h,d^2=ghi,e^2=gi,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dgh,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

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 396019);