#108032

order	512
nilpotency class	3
isoclinism	isoclinic class of #108026
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	C₂×#121	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

748 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,e^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gi,d^b=di,e^b=ehi,f^b=fh,d^c=dhi,e^c=ei,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, 15228);