#487202

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	C₂⁴	C₂²×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

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

normal subgroups only

This group has 681 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^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=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, 394398);