#484500

order	512
nilpotency class	4
isoclinism	isoclinic class of #484176
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	#537	C₂³
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

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

normal subgroups only

This group has 711 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) = 128

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

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

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=gi,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,e^c=eh,f^c=fi,f^e=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, 391696);