#110550

order	512
nilpotency class	3
isoclinism	isoclinic class of #110532
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1437
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#121	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

569 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,d^2=h,f^2=i,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=ghi,d^b=di,e^c=eh,f^c=fh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 17746);