#117278

order	512
nilpotency class	3
isoclinism	isoclinic class of #117269
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1439
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₂²⋊Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

514 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=i,d^2=h,f^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fh,c^b=cf,f^b=fhi,d^c=dh,f^c=fi,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 24474);