#107665

order	512
nilpotency class	3
isoclinism	isoclinic class of #107658
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	#3444	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

702 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,e^2=h,f^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=gi,d^b=di,e^b=eh,f^b=fh,d^c=dh,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, 14861);