#418480

order	512
nilpotency class	3
isoclinism	isoclinic class of #418480
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15263
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23414	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1631 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)(5,21)(6,25)(7,12)(8,16)(10,24)(11,27)(14,39)(15,35)(17,37)(18,32)(19,47)(20,34)(22,29)(23,36)(26,30)(28,48)(31,41)(33,49)(38,42)(40,60)(43,50)(44,55)(45,63)(46,51)(52,54)(53,56)(57,62)(58,61)(59,64);(1,3)(2,8)(4,14)(5,22)(6,26)(7,17)(9,31)(10,35)(11,37)(12,30)(13,25)(15,23)(16,27)(18,44)(19,48)(20,42)(21,50)(24,53)(28,34)(29,36)(32,58)(33,38)(39,47)(40,45)(41,49)(43,54)(46,62)(51,60)(52,56)(55,63)(57,59)(61,64);(1,4)(2,9)(3,14)(5,18)(6,19)(7,20)(8,28)(10,32)(11,33)(12,34)(13,38)(15,40)(16,41)(17,42)(21,51)(22,44)(23,45)(24,46)(25,47)(26,48)(27,49)(29,55)(30,31)(35,60)(36,59)(37,39)(43,61)(50,58)(52,63)(53,62)(54,64)(56,57);(1,5,6,23,7,24,27,54)(2,10,11,36,12,21,25,52)(3,15,16,22,17,43,26,53)(4,18,19,45,20,46,49,64)(8,29,13,35,30,56,37,50)(9,32,33,59,34,51,47,63)(14,40,41,44,42,61,48,62)(28,55,38,60,31,57,39,58)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319954, #72319969, #72320069
maximal normal subgroup of	#72319954, #72319969, #72320069

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 325676);