#398099

order	512
nilpotency class	3
isoclinism	isoclinic class of #396612
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2850
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14759
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1277 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)(3,9)(4,19,22,38)(5,25)(6,12)(7,28)(8,13)(10,23,37,24)(11,26)(14,35,42,58)(15,44)(16,32)(17,33)(18,34)(20,49,43,30)(21,51,52,36)(27,39)(29,40)(31,46)(41,55,61,64)(45,57)(47,56)(48,53,60,54)(50,59,62,63);(1,3)(2,9)(4,20,22,43)(5,15)(6,17)(7,16)(8,18)(10,35,37,58)(11,31)(12,33)(13,34)(14,24,42,23)(19,30,38,49)(21,50,52,62)(25,44)(26,46)(27,47)(28,32)(29,45)(36,59,51,63)(39,56)(40,57)(41,54,61,53)(48,55,60,64);(1,4,6,24,8,22,7,23)(2,10,12,38,13,37,28,19)(3,14,16,43,18,42,17,20)(5,21,26,54,27,52,29,53)(9,30,32,58,34,49,33,35)(11,36,39,60,40,51,25,48)(15,41,45,62,47,61,46,50)(31,55,44,63,57,64,56,59);(1,5)(2,11)(3,15)(4,21)(6,26)(7,29)(8,27)(9,31)(10,36)(12,39)(13,40)(14,41)(16,45)(17,46)(18,47)(19,48)(20,50)(22,52)(23,53)(24,54)(25,28)(30,55)(32,44)(33,56)(34,57)(35,59)(37,51)(38,60)(42,61)(43,62)(49,64)(58,63)

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=ef,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,f^b=fi,e^c=ei,f^c=fi,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, 305295);