#107728

order	512
nilpotency class	3
isoclinism	isoclinic class of #107718
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

744 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=h,e^2=h,f^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=ghi,d^b=di,e^b=ehi,d^c=dhi,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, 14924);