#487232

order	512
nilpotency class	4
isoclinism	isoclinic class of #486811
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

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

normal subgroups only

This group has 711 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) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

(1,2,6,11)(3,13,18,37)(4,9,22,35)(5,24,27,10)(7,12,14,28)(8,25,31,17)(15,39,41,63)(16,40,44,64)(19,46,47,61)(20,33,23,36)(21,50,52,34)(26,54,42,38)(29,56,57,43)(30,55,58,45)(32,51,59,48)(49,62,53,60);(1,3,14,41)(2,8,28,57)(4,19,20,44)(5,25,42,43)(6,18,7,15)(9,32,33,58)(10,37,54,39)(11,31,12,29)(13,38,63,24)(16,22,47,23)(17,26,56,27)(21,51,49,45)(30,35,59,36)(34,61,60,40)(46,62,64,50)(48,53,55,52);(3,15)(4,20)(5,26)(8,29)(9,33)(13,39)(16,44)(18,41)(19,47)(21,52)(22,23)(27,42)(30,58)(31,57)(32,59)(34,60)(35,36)(37,63)(40,64)(45,51)(46,61)(48,55)(49,53)(50,62);(1,4,7,23)(2,9,12,36)(3,16,15,19)(5,21,26,53)(6,22,14,20)(8,30,29,32)(10,34,38,62)(11,35,28,33)(13,40,39,46)(17,45,43,48)(18,44,41,47)(24,50,54,60)(25,55,56,51)(27,52,42,49)(31,58,57,59)(37,64,63,61)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,d^2=i,e^2=g,b^a=be,c^a=cf,e^a=eg,h^a=hi,c^b=ci,d^b=di,e^b=eg,f^b=fhi,d^c=dgi,e^c=eh,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, 394428);