#411979

order	512
nilpotency class	3
isoclinism	isoclinic class of #398772
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2439
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15125
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1083 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)(3,8)(4,17,20,9)(5,22)(6,25)(7,11)(10,23)(12,26)(13,34,18,27)(14,37)(15,29)(16,41)(19,46,47,50)(21,32,49,43)(24,51)(28,39)(30,40)(31,48,42,63)(33,52)(35,58,44,60)(36,54,45,57)(38,61)(53,59,56,64)(55,62);(1,3)(2,8)(4,18,20,13)(5,14)(6,16)(7,15)(9,27,17,34)(10,28)(11,29)(12,30)(19,44,47,35)(21,45,49,36)(22,37)(23,39)(24,38)(25,41)(26,40)(31,53,42,56)(32,54,43,57)(33,55)(46,58,50,60)(48,64,63,59)(51,61)(52,62);(1,4)(2,9)(3,13)(5,19)(6,21)(7,20)(8,27)(10,31)(11,17)(12,32)(14,35)(15,18)(16,36)(22,50)(23,48)(24,47)(25,43)(26,49)(28,53)(29,34)(30,54)(33,42)(37,60)(38,44)(39,59)(40,45)(41,57)(46,51)(52,63)(55,56)(58,61)(62,64);(1,5,6,23,7,24,26,52)(2,10,12,22,11,33,25,51)(3,14,16,39,15,38,40,62)(4,19,21,48,20,47,49,63)(8,28,30,37,29,55,41,61)(9,31,32,50,17,42,43,46)(13,35,36,59,18,44,45,64)(27,53,54,60,34,56,57,58)

Polycyclic presentation:

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

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, 319175);