#394482

order	512
nilpotency class	3
isoclinism	isoclinic class of #394482
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2435
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14717
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1910 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)(5,24)(6,12)(7,28)(8,13)(10,23,37,27)(11,25)(14,41,43,35)(15,45)(16,32)(17,33)(18,34)(20,30,44,49)(21,52,53,36)(26,39)(29,40)(31,47)(42,56,62,64)(46,58)(48,57)(50,54,60,55)(51,59,63,61);(1,3)(2,9)(4,20,22,44)(5,15)(6,17)(7,16)(8,18)(10,35,37,41)(11,31)(12,33)(13,34)(14,23,43,27)(19,49,38,30)(21,51,53,63)(24,45)(25,47)(26,48)(28,32)(29,46)(36,59,52,61)(39,57)(40,58)(42,54,62,55)(50,64,60,56);(1,4)(2,10)(3,14)(5,21)(6,27)(7,23)(8,22)(9,30)(11,36)(12,38)(13,37)(15,42)(16,20)(17,44)(18,43)(19,28)(24,50)(25,55)(26,53)(29,54)(31,56)(32,35)(33,41)(34,49)(39,60)(40,52)(45,59)(46,51)(47,63)(48,62)(57,61)(58,64);(1,5)(2,11)(3,15)(4,21)(6,25)(7,29)(8,26)(9,31)(10,36)(12,39)(13,40)(14,42)(16,46)(17,47)(18,48)(19,50)(20,51)(22,53)(23,54)(24,28)(27,55)(30,56)(32,45)(33,57)(34,58)(35,59)(37,52)(38,60)(41,61)(43,62)(44,63)(49,64)

Polycyclic presentation:

2^9|a^2=e,b^2=e,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,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, 301678);