#513193

order	512
nilpotency class	5
isoclinism	isoclinic class of #513193
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	D₄×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁴
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #443 ◁ #2170 ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

424 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:

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

Polycyclic presentation:

2^9|e^2=gh,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dgh,e^a=eg,f^a=fi,g^a=gh,h^a=hi,c^b=ci,d^b=de,e^b=eg,g^b=gh,h^b=hi,f^c=fi,e^d=eg,g^d=gh,h^d=hi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225555
maximal normal subgroup of	#72225555

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420389);