#501147

order	512
nilpotency class	4
isoclinism	isoclinic class of #501144
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#24404
commutator, Frattini, cosocle	C₂×#41	C₂⁴
prime radical	#23874	C₂
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×H(5,𝔽₂) ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

1046 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 1046 conjugacy classes of subgroups — too many to draw. Only its 138 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,e^b=ei,f^b=fhi,g^b=gi,e^c=eh,f^c=fi,g^c=gi,e^d=ei,f^d=fhi,f^e=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408343);