#399218

order	512
nilpotency class	3
isoclinism	isoclinic class of #393928
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2433
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14777
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1299 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=ef,c^2=ef,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,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, 306414);