#500356

order	512
nilpotency class	4
isoclinism	isoclinic class of #500355
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#24089
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#525	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2930 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

700 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250066, #72250076, #72281185, …
maximal normal subgroup of	#72250066, #72250076, #72281185, …
maximal quotient of	#72250661, #72280164, #72282164

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407552);