#107950

order	512
nilpotency class	3
isoclinism	isoclinic class of #107946
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:

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

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=eh,f^b=fh,d^c=dh,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, 15146);