#107619

order	512
nilpotency class	3
isoclinism	isoclinic class of #8491
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#379
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	C₂²⋊#95
prime radical	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

705 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=i,e^2=h,f^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,d^b=di,f^b=fhi,e^c=eh,f^c=fhi

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, 14815);