#107687

order	512
nilpotency class	3
isoclinism	isoclinic class of #107678
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	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

698 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=hi,e^2=h,f^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=gh,d^b=di,e^b=eh,f^b=fi,d^c=dh,e^c=eh,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, 14883);