#486936

order	512
nilpotency class	4
isoclinism	isoclinic class of #486803
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×C₂²⋊C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 746 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)(19,27)(20,24)(21,25)(22,26)(23,29)(28,41)(30,42)(31,45)(32,38)(33,39)(34,40)(35,43)(36,44)(37,46)(47,48);(2,6)(4,12)(7,13)(8,9)(11,16)(14,15)(17,19,22,30)(18,23,26,36)(20,31,33,28)(21,29,34,44)(24,37,39,35)(25,27,40,42)(32,46,47,43)(38,45,48,41);(1,3)(2,7)(4,11)(5,10)(6,13)(8,15)(9,14)(12,16)(21,34)(29,44)(32,47)(43,46);(3,10)(7,14)(11,16)(13,15)(17,20,22,33)(18,24,26,39)(19,28,30,31)(21,32,34,47)(23,35,36,37)(25,38,40,48)(27,41,42,45)(29,43,44,46)

Polycyclic presentation:

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

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72284971, #72284994, #72285303
maximal normal subgroup of	#72284971, #72284994, #72285303

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 394132);