#115539

order	512
nilpotency class	3
isoclinism	isoclinic class of #10120
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#392
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1438
prime radical	#60953	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

760 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)(3,12)(4,17)(5,22)(6,10)(7,11)(8,14)(9,20)(13,30)(15,25)(16,24)(18,29)(19,28)(21,27)(23,31)(26,32)(33,34,38,42,39,43,37,50)(35,40,45,55,46,56,51,54)(36,41,49,60,48,58,52,59)(44,53,62,57,61,64,47,63);(1,3)(2,8)(4,18)(5,15)(6,14)(7,16)(9,26)(10,24)(11,25)(12,22)(13,19)(17,23)(20,31)(21,32)(27,30)(28,29)(33,35)(34,40)(36,47)(37,51)(38,45)(39,46)(41,57)(42,55)(43,56)(44,49)(48,62)(50,54)(52,61)(53,59)(58,63)(60,64);(1,4)(2,9)(3,13)(5,19)(6,20)(7,21)(8,23)(10,27)(11,28)(12,29)(14,18)(15,32)(16,31)(17,22)(24,26)(25,30)(33,36)(34,41)(35,44)(37,52)(38,49)(39,48)(40,53)(42,60)(43,58)(45,62)(46,61)(47,51)(50,59)(54,63)(55,57)(56,64)

Polycyclic presentation:

2^9|a^2=g,d^2=h,e^2=h,f^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fi,c^b=cf,d^b=dh,f^b=fhi,d^c=di,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, 22735);