#491711

order	512
nilpotency class	4
isoclinism	isoclinic class of #491662
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#493
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	C₂³	#493
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

471 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:

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

Polycyclic presentation:

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

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, 398907);