#107212

order	512
nilpotency class	3
isoclinism	isoclinic class of #107194
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	C₂⁴	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#121	C₂×C₄
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

695 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

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

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, 14408);