#501155

order	512
nilpotency class	4
isoclinism	isoclinic class of #501150
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	#23875	C₂
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×2₋¹⁺⁴ ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

1040 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 1040 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^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,d^c=di,e^c=eh,f^c=fi,g^c=gi,e^d=ei,f^d=fhi,f^e=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408351);