#405292

order	512
nilpotency class	3
isoclinism	isoclinic class of #395587
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2411
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14911
prime radical	#60387	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2168 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)(5,11)(6,13)(7,12)(8,14)(9,18)(10,24)(16,30)(17,41)(19,38)(21,50)(22,52)(23,35)(25,34)(26,39)(27,37)(28,32)(29,36)(31,44)(33,53)(40,43)(42,61)(45,58)(46,57)(47,48)(49,54)(51,56)(55,59)(60,63)(62,64);(1,3)(2,9)(4,21,25,16)(5,17)(6,28)(7,18)(8,19)(10,30,35,50)(11,31)(12,38)(13,15)(14,32)(20,48,34,40)(22,51,54,42)(23,43,24,47)(26,45)(27,46)(29,44)(33,56,59,61)(36,57)(37,58)(39,41)(49,64,52,60)(53,62,55,63);(1,4)(2,10)(3,16)(5,22)(6,24)(7,23)(8,25)(9,30)(11,33)(12,20)(13,34)(14,35)(15,40)(17,42)(18,47)(19,21)(26,55)(27,54)(28,43)(29,53)(31,56)(32,50)(36,49)(37,59)(38,48)(39,52)(41,60)(44,63)(45,62)(46,51)(57,64)(58,61);(1,5)(2,11)(3,17)(4,22)(6,26)(7,29)(8,27)(9,31)(10,33)(12,36)(13,39)(14,37)(15,41)(16,42)(18,44)(19,46)(20,49)(21,51)(23,53)(24,55)(25,54)(28,45)(30,56)(32,58)(34,52)(35,59)(38,57)(40,60)(43,62)(47,63)(48,64)(50,61)

Polycyclic presentation:

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

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, 312488);