#454685

order	512
nilpotency class	3
isoclinism	isoclinic class of #454109
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2670
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16255
prime radical	#60366	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1763 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)(4,21)(5,27,31,11)(6,13)(7,12)(8,14)(9,18)(10,25)(16,45)(17,44,52,59)(19,35)(20,34)(22,43)(23,40,54,56)(24,39)(26,38)(28,37,53,61)(29,41,30,42)(32,48)(33,50,55,51)(36,47)(46,57,64,62)(49,60,63,58);(1,3)(2,9)(4,22)(5,17)(6,19)(7,18)(8,20)(10,36)(11,33)(12,34)(13,15)(14,35)(16,25)(21,45)(23,49)(24,48)(26,47)(27,55)(28,46)(29,51)(30,50)(31,52)(32,38)(37,60)(39,43)(40,62)(41,44)(42,59)(53,64)(54,63)(56,57)(58,61);(1,4)(2,10)(3,16)(5,28)(6,25)(7,24)(8,26)(9,32)(11,40)(12,21)(13,38)(14,39)(15,43)(17,49)(18,22)(19,47)(20,48)(23,29)(27,56)(30,54)(31,53)(33,58)(34,36)(35,45)(37,41)(42,61)(44,62)(46,50)(51,64)(52,63)(55,60)(57,59);(1,5)(2,11)(3,17)(4,23)(6,30)(7,29)(8,31)(9,33)(10,37)(12,42)(13,41)(14,27)(15,44)(16,46)(18,51)(19,50)(20,52)(21,40)(22,49)(24,53)(25,28)(26,54)(32,57)(34,59)(35,55)(36,60)(38,56)(39,61)(43,58)(45,62)(47,63)(48,64)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 361881);