#317273

order	512
nilpotency class	3
isoclinism	isoclinic class of #317267
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2535
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#532	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

456 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)(3,11)(4,15)(5,18)(6,10)(7,13)(8,17)(9,19)(12,25)(14,23)(16,29)(20,28)(21,26)(22,30)(24,27)(31,32)(33,34)(35,40)(36,48)(37,53)(38,42)(39,43)(41,50)(44,57)(45,62)(46,55)(47,56)(49,63)(51,58)(52,59)(54,60)(61,64);(1,3)(2,7,10,20)(4,12)(5,13)(6,14)(8,21,22,25)(9,11,18,23)(15,24,29,31)(16,26)(17,27)(19,28)(30,32)(33,35,38,46,39,47,37,45)(34,40,42,55,43,56,53,62)(36,49,51,61,52,60,50,44)(41,57,48,63,58,64,59,54);(4,16)(7,20)(8,22)(11,23)(12,26)(15,29)(17,30)(27,32)(33,36,39,52)(34,41,43,58)(35,44,47,61)(37,51,38,50)(40,54,56,63)(42,59,53,48)(45,49,46,60)(55,64,62,57);(1,4,5,17,6,16,19,30)(2,8,9,15,10,22,18,29)(3,12,13,27,14,26,28,32)(7,21,11,24,20,25,23,31)

Polycyclic presentation:

2^9|b^2=g,c^2=i,d^2=e,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=gh,c^b=cg,f^b=fh,d^c=dh,f^c=fi,g^c=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, 224469);