#487029

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
prime radical	#53977	C₂
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×C₄⋊C₄ ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 742 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:

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

Polycyclic presentation:

2^9|a^2=g,c^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, 394225);