#487118

order	512
nilpotency class	4
isoclinism	isoclinic class of #486859
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×C₄⋊C₄	C₂⁴
prime radical	C₂×#537	C₂²
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×C₄⋊C₄ ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

783 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 783 conjugacy classes of subgroups — too many to draw. Only its 118 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:

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

Polycyclic presentation:

2^9|b^2=gi,c^2=i,e^2=g,f^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fi,h^a=hi,e^b=eg,f^b=fhi,d^c=dgi,e^c=eh,f^c=fi,f^e=fi

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 394314);