#499824

order	512
nilpotency class	4
isoclinism	isoclinic class of #499819
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#23900
commutator, Frattini, cosocle	C₂×#41	C₂⁴
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×H(5,𝔽₂) ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

1864 conjugacy classes of subgroups.

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72281946, #72282045, #72282105, …
maximal normal subgroup of	#72281946, #72282045, #72282105, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407020);