#501111

order	512
nilpotency class	4
isoclinism	isoclinic class of #501108
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

1045 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 1045 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)(3,14)(4,9)(5,11,25,10)(6,12)(7,13)(8,16,31,23)(15,44)(17,45)(18,39)(19,51)(20,35,24,34)(21,36)(22,37)(26,40,57,38)(27,58)(28,42)(29,43)(30,46,59,48)(32,49,41,55)(33,52,60,53)(47,64)(50,62)(54,63,56,61);(1,3,7,18)(2,8,13,31)(4,19,22,50)(5,23,25,16)(6,27,29,17)(9,33,37,60)(10,14,11,39)(12,41,43,32)(15,28,47,21)(20,53,24,52)(26,49,57,55)(30,42,59,36)(34,51,35,62)(38,58,40,45)(44,63,64,61)(46,54,48,56);(1,4,7,22)(2,9,13,37)(3,15,18,47)(5,24,25,20)(6,28,29,21)(8,30,31,59)(10,34,11,35)(12,42,43,36)(14,44,39,64)(16,48,23,46)(17,50,27,19)(26,54,57,56)(32,60,41,33)(38,63,40,61)(45,62,58,51)(49,52,55,53);(2,10)(5,25)(8,14)(9,34)(11,13)(12,38)(16,23)(20,24)(26,57)(30,44)(31,39)(32,45)(33,51)(35,37)(36,61)(40,43)(41,58)(42,63)(46,48)(49,55)(52,53)(54,56)(59,64)(60,62)

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72341696, #72341943, #72342030, …
maximal normal subgroup of	#72341696, #72341943, #72342030, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408307);