#488274

order	512
nilpotency class	4
isoclinism	isoclinic class of #487707
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	#3497	C₂²
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

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

normal subgroups only

This group has 742 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=gi,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,d^b=di,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, 395470);