#454671

order	512
nilpotency class	3
isoclinism	isoclinic class of #454670
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#16255
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1752 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318217, #72318925, #72319003
maximal normal subgroup of	#72318217, #72318925, #72319003

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 361867);