#107277

order	512
nilpotency class	3
isoclinism	isoclinic class of #107262
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²⋊#95
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

686 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)(3,13)(4,18,21,26)(5,11)(6,10)(7,12)(8,15)(9,20,25,19)(14,28,31,22)(16,24)(17,23)(27,30,32,29)(33,34,38,42,39,44,37,43)(35,45,47,40,48,55,53,56)(36,49,51,41,52,57,50,58)(46,60,61,64,62,54,63,59);(1,3,6,16,7,17,5,15)(2,8,11,23,12,24,10,13)(4,14,20,30,21,31,19,29)(9,22,26,32,25,28,18,27)(33,35)(34,40)(36,46)(37,53)(38,47)(39,48)(41,54)(42,55)(43,45)(44,56)(49,64)(50,63)(51,61)(52,62)(57,59)(58,60);(1,4,7,21)(2,9,12,25)(3,14,17,31)(5,19,6,20)(8,22,24,28)(10,18,11,26)(13,27,23,32)(15,29,16,30)(33,36,39,52)(34,41,44,58)(35,46,48,62)(37,51,38,50)(40,54,56,60)(42,49,43,57)(45,59,55,64)(47,63,53,61)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=h,d^2=h,e^2=h,f^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,g^a=ghi,d^b=di,e^b=ei,d^c=di,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, 14473);