#486904

order	512
nilpotency class	4
isoclinism	isoclinic class of #486867
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
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

825 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 825 conjugacy classes of subgroups — too many to draw. Only its 116 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)(3,7)(4,6)(5,9)(8,12)(10,14)(11,13)(15,16)(17,18)(19,23)(20,25)(21,24)(22,27)(26,35)(28,38)(29,37)(30,39)(31,41)(32,40)(33,43)(34,36)(42,46)(44,48)(45,47);(2,6)(4,12)(7,13)(8,9)(11,16)(14,15)(17,19,22,28)(18,23,27,38)(20,31,33,29)(21,34,35,30)(24,39,26,36)(25,41,43,37)(32,44,46,45)(40,47,42,48);(1,3)(2,7)(4,11)(5,10)(6,13)(8,15)(9,14)(12,16)(18,24)(19,28)(21,35)(23,36)(25,40)(26,27)(29,31)(32,46)(37,47)(38,39)(41,48)(42,43);(3,10)(7,14)(11,16)(13,15)(17,20,22,33)(18,25,27,43)(19,29,28,31)(21,32,35,46)(23,37,38,41)(24,40,26,42)(30,44,34,45)(36,47,39,48)

Polycyclic presentation:

2^9|b^2=i,d^2=i,e^2=g,f^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fi,h^a=hi,c^b=ci,d^b=di,e^b=eg,f^b=fhi,d^c=dg,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, 394100);