#490399

order	512
nilpotency class	4
isoclinism	isoclinic class of #490107
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	#62379	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

647 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)(35,40)(36,39)(37,52)(38,44)(41,58)(42,49)(43,50)(45,60)(46,53)(47,64)(48,57)(51,54)(55,63)(56,62)(59,61);(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)(34,39)(36,49)(37,50)(40,53)(41,52)(42,43)(44,54)(45,61)(46,63)(47,62)(51,58)(55,56)(57,60)(59,64);(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)(34,40)(36,46)(37,47)(38,48)(39,53)(41,59)(42,55)(43,56)(44,57)(45,51)(49,63)(50,62)(52,64)(54,60)(58,61);(3,15)(8,23)(13,24)(14,29)(16,17)(22,28)(27,32)(30,31)(33,36,50,58,38,51,37,49)(34,41,52,39,44,42,43,54)(35,45,62,63,48,46,47,61)(40,55,64,60,57,59,56,53)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 397595);