#494169

order	512
nilpotency class	4
isoclinism	isoclinic class of #483412
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#427
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#1846
prime radical	C₂⁴	#135
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

621 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)(3,14,16,37)(4,20,23,50)(5,25)(6,12)(7,29)(8,13)(9,17,31,44)(10,22,34,24)(11,27)(15,39)(18,38,19,47)(21,51,53,33)(26,35)(28,55)(30,57)(32,45,56,46)(36,43)(40,60)(41,62)(42,59)(48,61)(49,52,64,54)(58,63);(1,3,5,17,7,19,27,46,8,16,28,44,6,18,26,45)(2,9,11,14,12,32,25,38,13,31,35,37,29,56,55,47)(4,15,21,41,24,43,54,63,23,40,53,61,22,42,52,57)(10,30,33,36,20,48,49,39,34,58,51,59,50,62,64,60);(1,4)(2,10,13,34)(3,15)(5,21)(6,22)(7,24)(8,23)(9,30,31,58)(11,33,35,51)(12,20,29,50)(14,36,37,59)(16,40)(17,41)(18,42)(19,43)(25,49,55,64)(26,52)(27,54)(28,53)(32,48,56,62)(38,39,47,60)(44,61)(45,57)(46,63);(3,16)(9,31)(14,37)(15,40)(17,44)(18,19)(30,58)(32,56)(36,59)(38,47)(39,60)(41,61)(42,43)(45,46)(48,62)(57,63)

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=fh,e^2=h,f^2=i,h^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,d^b=di,g^b=gi,g^c=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401365);