#108016

order	512
nilpotency class	3
isoclinism	isoclinic class of #108006
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	C₂×#121	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

754 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=h,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=ghi,d^b=di,e^b=ehi,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, 15212);