#110744

order	512
nilpotency class	3
isoclinism	isoclinic class of #9331
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1437
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

546 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=efh,d^2=i,f^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,d^b=di,e^b=ehi,f^b=fhi,d^c=dhi,e^c=eh,f^c=fh

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, 17940);