#394687

order	512
nilpotency class	3
isoclinism	isoclinic class of #394481
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2472
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14718
prime radical	#60387	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2104 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)(6,13)(7,12)(8,14)(9,18)(10,24)(11,28)(16,31)(17,32)(19,34)(20,33)(22,35)(23,52)(25,38)(26,37)(29,39)(30,40)(36,54)(41,44)(42,47)(43,59)(45,49)(46,56)(48,58)(50,53)(51,64)(55,60)(57,62)(61,63);(1,3)(2,9)(4,22)(5,17)(6,19)(7,18)(8,20)(10,35)(11,32)(12,33)(13,15)(14,34)(16,24)(21,49)(23,51)(25,45)(26,44)(27,56)(28,47)(29,48)(30,46)(31,37)(36,59)(38,41)(39,42)(40,58)(43,53)(50,64)(52,61)(54,62)(55,63)(57,60);(1,4,6,24,8,26,7,25)(2,10,12,21,14,38,13,37)(3,16,18,22,20,45,19,44)(5,23,28,53,29,54,30,55)(9,31,15,41,34,49,33,35)(11,36,27,50,40,52,39,60)(17,43,46,51,48,63,47,62)(32,57,42,61,58,64,56,59);(1,5)(2,11)(3,17)(4,23)(6,28)(7,30)(8,29)(9,32)(10,36)(12,27)(13,39)(14,40)(15,42)(16,43)(18,46)(19,47)(20,48)(21,50)(22,51)(24,53)(25,55)(26,54)(31,57)(33,56)(34,58)(35,59)(37,60)(38,52)(41,61)(44,62)(45,63)(49,64)

Polycyclic presentation:

2^9|c^2=e,e^2=i,f^2=i,g^2=i,h^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,g^b=gi,h^d=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 301883);