#394682

order	512
nilpotency class	3
isoclinism	isoclinic class of #394483
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

2094 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)(4,20)(5,26)(6,14)(7,12)(8,13)(9,18)(10,24)(11,27)(16,40)(17,32)(19,38)(21,49)(22,52)(23,35)(25,36)(28,39)(29,33)(30,37)(31,43)(34,55)(41,46)(42,58)(44,56)(45,59)(47,51)(48,53)(50,63)(54,57)(60,62)(61,64);(1,3)(2,9)(4,21)(5,17)(6,29)(7,18)(8,19)(10,31)(11,32)(12,38)(13,33)(14,15)(16,24)(20,47)(22,50)(23,43)(25,51)(26,56)(27,46)(28,45)(30,44)(34,58)(35,49)(36,40)(37,59)(39,41)(42,54)(48,63)(52,60)(53,61)(55,62)(57,64);(1,4,6,23,8,24,7,25)(2,10,12,35,13,20,14,36)(3,16,18,43,19,21,29,51)(5,22,27,53,28,54,30,55)(9,31,15,40,33,47,38,49)(11,34,26,57,37,48,39,52)(17,42,44,61,45,50,46,62)(32,58,41,60,59,63,56,64);(1,5)(2,11)(3,17)(4,22)(6,27)(7,30)(8,28)(9,32)(10,34)(12,26)(13,37)(14,39)(15,41)(16,42)(18,44)(19,45)(20,48)(21,50)(23,53)(24,54)(25,55)(29,46)(31,58)(33,59)(35,57)(36,52)(38,56)(40,60)(43,61)(47,63)(49,64)(51,62)

Polycyclic presentation:

2^9|c^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,f^c=fi,g^c=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, 301878);