#464073

order	512
nilpotency class	3
isoclinism	isoclinic class of #464072
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#16500
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23412	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1415 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)(3,13)(4,9)(5,10)(6,24)(7,12)(8,16)(11,27)(14,40)(15,46)(17,42)(18,32)(19,38)(20,34)(21,55)(22,39)(23,37)(25,33)(26,36)(28,50)(29,48)(30,52)(31,44)(35,54)(41,49)(43,62)(45,57)(47,60)(51,59)(53,61)(56,64)(58,63);(1,3)(2,8,12,30)(4,14)(5,15)(6,16)(7,17)(9,31,34,28)(10,29,37,59)(11,13,24,42)(18,43)(19,44)(20,45)(21,47)(22,48)(23,49)(25,50)(26,51)(27,52)(32,60,61,56)(33,40,38,57)(35,62,55,58)(36,46,39,41)(53,63)(54,64);(1,4,7,20)(2,9,12,34)(3,14,17,45)(5,21,23,54)(6,25,27,19)(8,28,30,31)(10,35,37,55)(11,38,24,33)(13,40,42,57)(15,47,49,64)(16,50,52,44)(18,26,53,22)(29,58,59,62)(32,39,61,36)(41,60,46,56)(43,51,63,48);(1,5)(2,10)(3,15)(4,18)(6,26)(7,23)(8,29)(9,32)(11,39)(12,37)(13,41)(14,43)(16,51)(17,49)(19,54)(20,53)(21,25)(22,27)(24,36)(28,56)(30,59)(31,60)(33,55)(34,61)(35,38)(40,58)(42,46)(44,64)(45,63)(47,50)(48,52)(57,62)

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72264914, #72264928, #72265047
maximal normal subgroup of	#72264914, #72264928, #72265047
maximal quotient of	#72265539

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 371269);