#142622

order	512
nilpotency class	4
isoclinism	isoclinic class of #142620
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#9946
commutator, Frattini, cosocle	C₂×#131	C₂³
prime radical	#25799	C₂
LCS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁

563 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 563 conjugacy classes of subgroups — too many to draw. Only its 64 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:

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

Polycyclic presentation:

2^9|a^2=i,d^2=gi,e^2=gi,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,f^a=fgh,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=eg,f^b=fi,g^b=gi,d^c=dh,e^c=eg,f^c=fi,g^c=gi,h^c=hi,f^d=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, 49818);