#488005

order	512
nilpotency class	4
isoclinism	isoclinic class of #487691
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×Q₈ ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

731 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 731 conjugacy classes of subgroups — too many to draw. Only its 120 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:

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

Polycyclic presentation:

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

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 395201);