#151151

order	512
nilpotency class	5
isoclinism	isoclinic class of #151150
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#12090
commutator, Frattini, cosocle	#408	C₂³
LCS	G ▷ #408 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #408 ◁ G
LECS	G ▷ #408 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

563 conjugacy classes of subgroups; only the 38 normal subgroups are shown.

normal subgroups only

This group has 563 conjugacy classes of subgroups — too many to draw. Only its 38 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 — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|c^2=i,d^2=fi,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=df,f^a=fi,g^a=gh,d^b=df,e^b=eghi,f^b=fi,h^b=hi,d^c=dg,g^c=gi,h^c=hi,e^d=ehi,g^d=gi,g^e=gi

Conjugacy classes & character table

35 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 58347);