#108377

order	512
nilpotency class	3
isoclinism	isoclinic class of #108372
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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=hi,e^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=gi,d^b=dh,e^b=eh,f^b=fi,d^c=dh,e^c=ei

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, 15573);