#110739

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	#117
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

534 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)(3,11,13,7)(4,8,15,20)(5,10)(6,9)(12,18)(14,24,27,19)(16,21,28,30)(17,22)(23,25)(26,29)(31,32)(33,34,38,42,39,43,37,50)(35,44,46,40,47,55,51,54)(36,41,48,57,49,58,52,56)(45,59,62,63,61,64,60,53);(1,3,5,14,6,13,17,27)(2,7,10,19,9,11,22,24)(4,12,16,26,15,25,28,32)(8,18,21,29,20,23,30,31)(33,35)(34,40)(36,45)(37,51)(38,46)(39,47)(41,53)(42,55)(43,54)(44,50)(48,62)(49,61)(52,60)(56,64)(57,59)(58,63);(1,4,6,15)(2,8)(3,12,13,25)(5,16,17,28)(7,18)(9,20)(10,21)(11,23)(14,26,27,32)(19,29)(22,30)(24,31)(33,36,37,48,39,49,38,52)(34,41,42,56,43,58,50,57)(35,45,51,62,47,61,46,60)(40,53,55,64,54,63,44,59)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=efh,d^2=i,f^2=i,g^2=hi,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, 17935);