#107397

order	512
nilpotency class	3
isoclinism	isoclinic class of #107386
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	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

720 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 14593);