#490412

order	512
nilpotency class	4
isoclinism	isoclinic class of #490121
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#62384	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

644 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=h,c^2=h,d^2=ghi,e^2=gi,f^2=hi,g^2=i,b^a=be,c^a=cf,d^a=dghi,e^a=eg,f^a=fhi,g^a=gi,d^b=dg,e^b=eg,f^b=fh,g^b=gi,e^c=eh,f^c=fhi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 397608);