#488539

order	512
nilpotency class	4
isoclinism	isoclinic class of #488539
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
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#408 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 895 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) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 395735);