#506306

order	512
nilpotency class	4
isoclinism	isoclinic class of #506304
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#28165
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#33595	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1747 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

548 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)(4,18)(5,6)(7,28)(8,25)(10,21)(11,31)(12,27)(13,41)(14,34)(15,46)(16,47)(17,23)(19,53)(20,52)(22,56)(24,55)(26,51)(29,58)(30,40)(32,39)(33,43)(35,49)(36,54)(37,57)(38,60)(42,48)(44,64)(45,63)(50,62)(59,61);(1,3)(2,9,5,23)(4,13)(6,17)(7,29)(8,16)(10,36,19,33)(11,38,24,35)(12,14,25,47)(15,26)(18,41,52,63)(20,45)(21,54)(22,44)(27,34)(28,58,51,46)(30,59)(31,60)(32,50)(37,42,40,61)(39,62,56,64)(43,53)(48,57)(49,55);(1,4,6,20)(2,10,5,19)(3,13,17,45)(7,30,26,57)(8,32,27,22)(9,33,23,36)(11,39,24,56)(12,40,25,37)(14,42,47,61)(15,48,29,59)(16,50,34,44)(18,51,52,28)(21,55,53,31)(35,62,38,64)(41,58,63,46)(43,60,54,49);(1,5,6,2)(3,14)(4,19,20,10)(7,24,26,11)(8,25,27,12)(9,34)(13,42)(15,46)(16,23)(17,47)(18,21,52,53)(22,40,32,37)(28,31,51,55)(29,58)(30,56,57,39)(33,44)(35,60)(36,50)(38,49)(41,48)(43,64)(45,61)(54,62)(59,63)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 413502);