#500167

order	512
nilpotency class	4
isoclinism	isoclinic class of #500164
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#24033
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#61053	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2929 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

938 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307079, #72307097, #72307112, …
maximal normal subgroup of	#72307079, #72307097, #72307112, …
maximal quotient of	C₂×#500167

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407363);