#500647

order	512
nilpotency class	4
isoclinism	isoclinic class of #500646
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#24196
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#13334	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2960 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

756 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)(3,14)(4,9)(5,24)(6,29)(7,13)(8,15)(10,25)(11,40)(12,27)(16,48)(17,31)(18,42)(19,52)(20,56)(21,59)(22,61)(23,38)(26,44)(28,41)(30,39)(32,50)(33,47)(34,51)(35,55)(36,57)(37,58)(43,60)(45,49)(46,62)(53,63)(54,64);(3,15)(5,25)(6,27)(8,31)(10,24)(11,41)(12,29)(14,33)(16,46)(17,47)(18,51)(19,53)(20,57)(21,60)(22,58)(26,30)(28,40)(32,48)(34,39)(35,52)(36,56)(37,61)(42,44)(43,59)(45,62)(49,50)(54,55)(63,64);(1,3)(2,8)(4,19)(5,26)(6,30)(7,17)(9,35)(10,39)(11,42)(12,44)(13,33)(14,24)(15,25)(16,22)(18,27)(20,50)(21,46)(23,54)(28,51)(29,34)(31,41)(32,37)(36,48)(38,63)(40,47)(43,45)(49,58)(52,56)(53,60)(55,57)(59,64)(61,62);(1,4,7,23)(2,9,13,38)(3,16)(5,20,28,21)(6,22,27,58)(8,32)(10,36,11,43)(12,37,29,61)(14,45)(15,46)(17,49)(18,19)(24,56,41,59)(25,57,40,60)(26,55)(30,54)(31,48)(33,62)(34,35)(39,52)(42,64)(44,63)(47,50)(51,53)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250066
maximal normal subgroup of	#72250066
maximal quotient of	#72250663, #72281717, #72282014

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407843);