#110149

order	512
nilpotency class	3
isoclinism	isoclinic class of #110146
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	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

535 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)(35,43)(36,47,49,55)(37,41)(38,42)(39,45)(40,48,54,50)(44,59,60,64)(46,58)(51,56)(52,63,57,61)(53,62);(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)(34,39,42,43)(36,44)(37,46)(38,45)(40,52,54,57)(41,53,56,58)(47,64,55,59)(48,61)(49,60)(50,63)(51,62);(1,4)(2,8,9,20)(3,12)(5,16)(6,15)(7,18,11,23)(10,21,22,30)(13,25)(14,26)(17,28)(19,29,24,31)(27,32)(33,36,37,50,38,49,51,48)(34,40,41,55,42,54,56,47)(35,44,46,63,45,60,62,61)(39,52,53,59,43,57,58,64)

Polycyclic presentation:

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

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, 17345);